Model Comparison: Algebra

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Gemini 3 Pro

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What You Need to Know

• Algebra is the Heavy Lifter: Algebra (including Pre-Algebra, Elementary, and Intermediate Algebra) makes up a significant portion of the ACT Math section. Mastery here is critical for a high score.

• Translation is Key: A huge number of questions test your ability to translate word problems into mathematical expressions or equations (Modeling).

• Manipulation Matters: You must be fluent in operations with polynomials, radicals, and exponents to simplify complex expressions quickly.

• Know Your Forms: Recognizing the standard forms of linear (y=mx+b) and quadratic (ax^2+bx+c=0) equations instantly reveals properties like slope, intercepts, and roots.

Linear Equations and Inequalities

Linear relationships form the foundation of the ACT Algebra section. You must be comfortable solving for a specific variable and interpreting equations in real-world contexts.

Solving and Modeling

Most linear equations require isolating a variable using inverse operations. When modeling word problems, identify the constant (y-intercept) and the rate of change (slope).

• Slope-Intercept Form: y = mx + b

  • m is the slope (rate of change).

  • b is the y-intercept (starting value when x=0).

• Point-Slope Form: y - y1 = m(x - x1)

  • Useful when writing an equation given a point and a slope.

Inequalities Rule: When solving inequalities, you treat them exactly like equations, with one major exception: if you multiply or divide both sides by a negative number, you must flip the direction of the inequality symbol.

-3x > 12 \implies x < -4

Exam Focus

• Why it matters: Linear modeling is the most common way the ACT tests "real-world" math.

• Typical question patterns:

  • "Which equation represents the total cost C if a plumber charges 50 dollars to visit and 80 dollars per hour?"

  • Solving for a variable in terms of other variables (Literal Equations): "Solve A = p + prt for r."

• Common mistakes: Forgetting to flip the inequality sign when dividing by a negative; mixing up the slope (rate) with the y-intercept (flat fee) in word problems.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution is the point where the lines intersect.

Solving Methods

1. Substitution: Isolate one variable in the first equation and plug it into the second.

2. Elimination: Add or subtract the equations to cancel out one variable.

Special Case Solutions

Not all systems have exactly one solution (x, y). You must recognize these conditions:

• One Solution: The lines have different slopes. They intersect at one point.

• No Solution: The lines are parallel. They have the same slope but different y-intercepts. (m1 = m2, b1 \neq b2).

• Infinite Solutions: The lines are identical. They have the same slope and the same y-intercept. (m1 = m2, b1 = b2).

Exam Focus

• Why it matters: Systems test your ability to coordinate multiple constraints simultaneously.

• Typical question patterns:

  • Solving for x and y, then finding the value of an expression like x - y.

  • "For what value of k does the system of equations have no solution?" (Hint: Set the slopes equal to each other).

• Common mistakes: Finding x and stopping without finding y; forgetting that "no solution" implies parallel lines.

Polynomial Expressions and Equations

Polynomials involve sums of terms containing variables raised to non-negative integer powers. The ACT tests operations (add, subtract, multiply, divide) and factorization.

Operations

• Combining Like Terms: You can only add/subtract terms with the exact same variable and exponent (e.g., 3x^2 + 5x^2 = 8x^2).

• Multiplying Binomials (FOIL): First, Outer, Inner, Last.
  (x + 3)(x - 2) = x^2 - 2x + 3x - 6 = x^2 + x - 6

Important Identities

Memorizing these "Special Products" saves valuable time:

• Difference of Squares: a^2 - b^2 = (a - b)(a + b)

• Perfect Square Trinomials:

  • (a + b)^2 = a^2 + 2ab + b^2

  • (a - b)^2 = a^2 - 2ab + b^2

Exam Focus

• Why it matters: Polynomial manipulation is the "grammar" of algebra needed for advanced topics.

• Typical question patterns:

  • "Which of the following is equivalent to…" usually requiring FOIL or factoring.

  • Simplifying rational expressions where you must factor the numerator and denominator to cancel terms.

• Common mistakes: Distributing exponents incorrectly, such as thinking (x+3)^2 = x^2 + 9 (It does NOT! You must FOIL).

Quadratic Equations and Factoring

Quadratics appear frequently, asking for roots (solutions), vertices, or factors. The standard form is ax^2 + bx + c = 0.

Solving Quadratics

1. Factoring: Find two numbers that multiply to ac and add to b, then set factors to zero.
   x^2 - 5x + 6 = 0 \implies (x-2)(x-3)=0 \implies x=2, x=3

2. Quadratic Formula: If it doesn't factor easily, use:
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The Discriminant

The part under the radical, b^2 - 4ac, is called the discriminant. It tells you the nature of the roots:

• If b^2 - 4ac > 0: Two distinct real solutions.

• If b^2 - 4ac = 0: One real solution (a "double root").

• If b^2 - 4ac < 0: Two complex (imaginary) solutions.

Exam Focus

• Why it matters: Quadratics are the most complex algebraic equations regularly tested on the ACT.

• Typical question patterns:

  • "What are the solutions to the equation…"

  • "How many real solutions does the equation have?"

  • Finding the x-intercepts of a parabola (which is the same as solving for y=0).

• Common mistakes: Forgetting the \,\pm\, sign when taking the square root of both sides; calculating the discriminant incorrectly due to sign errors with a, b, or c.

Radical Expressions and Equations

Radicals involve roots (\sqrt{x}). You must be able to convert between radical form and rational exponent form.

Rules and Conversions

• Rational Exponents: x^{\frac{a}{b}} = \sqrt[b]{x^a}. The numerator is the power; the denominator is the root.

  • Example: 8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4

• Simplifying Radicals: Break the number inside the root into perfect square factors.

  • \sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}

Solving Radical Equations

To solve \sqrt{x+2} = x-4, square both sides. Crucial Step: You MUST check for extraneous solutions. Squaring both sides can introduce false answers that work algebraically but not in the original equation.

Exam Focus

• Why it matters: Rational exponents are often used to confuse students who only know integer powers.

• Typical question patterns:

  • "For all x > 0, which of the following is equivalent to \sqrt[3]{x^5}?"

  • Simplifying an expression involving sums of radicals like 3\sqrt{8} + 5\sqrt{18}.

• Common mistakes: Adding radicals that aren't like terms (e.g., \sqrt{2} + \sqrt{3} \neq \sqrt{5}); failing to check for extraneous solutions.

Exponential Equations and Growth/Decay

Exponential functions model rapid growth or decay. The standard form is y = ab^x, where a is the initial value and b is the growth factor.

Laws of Exponents

Mastery of these rules is non-negotiable:

1. Product Rule: x^a \cdot x^b = x^{a+b}

2. Quotient Rule: \frac{x^a}{x^b} = x^{a-b}

3. Power Rule: (x^a)^b = x^{ab}

4. Negative Exponents: x^{-n} = \frac{1}{x^n} (Flip the fraction, make the exponent positive).

5. Zero Exponent: x^0 = 1 (for x \neq 0).

Modeling Growth and Decay

• Growth: b = 1 + r (where r is the rate).

• Decay: b = 1 - r.

• Example: A population starts at 100 and grows by 5\% per year. The equation is P = 100(1.05)^t.

Exam Focus

• Why it matters: Exponents are used to test abstract arithmetic rules and financial/population modeling.

• Typical question patterns:

  • Simplifying messy fractions with variables and negative exponents.

  • "If 3^x = 81, what is x?" (Solve by making bases the same: 3^x = 3^4).

• Common mistakes: Multiplying bases together (2^3 \cdot 2^4 \neq 4^7, it is 2^7); treating negative exponents as negative numbers (they represent reciprocals).

Graphing and Modeling Expressions

This topic bridges Algebra and Coordinate Geometry. You must relate the algebraic equation to its visual graph.

Translations and Shifts

Given a function y = f(x):

• y = f(x) + k shifts the graph up k units.

• y = f(x - h) shifts the graph right h units. (Note: x-h moves right, x+h moves left).

• y = -f(x) reflects the graph over the x-axis.

Interpreting Models

If given a model C = 120 + 0.15m for the cost of a car rental:

• The 120 represents the y-intercept (fixed starting cost).

• The 0.15 represents the slope (cost per mile).

Exam Focus

• Why it matters: The ACT emphasizes "Mathematical Models"—connecting math to real-world scenarios.

• Typical question patterns:

  • "The graph of y = (x-3)^2 + 4 is shifted 2 units down. What is the new equation?"

  • Matching a word problem description to the correct graph shape.

• Common mistakes: Confusing horizontal shifts (inside the parentheses) with vertical shifts (outside); getting the direction of the horizontal shift wrong (x+2 is Left, not Right).

Quick Review Checklist

• Can you calculate slope given two points (x1, y1) and (x2, y2)?

• Do you know when to flip the inequality sign?

• Can you factor a difference of squares like x^2 - 16?

• Can you memorize the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}?

• Do you know that x^{-2} means \frac{1}{x^2}?

• Can you convert \sqrt[3]{x^2} to x^{2/3}?

• Can you solve a system of equations using elimination?

• Do you know that parallel lines have equal slopes?

• Can you expand (x+y)^2 without missing the middle term?

Final Exam Pitfalls

1. The "Freshman Dream" Error: Students often write (a+b)^2 = a^2 + b^2. This is WRONG. You must include the middle term: a^2 + 2ab + b^2.

2. Dividing Inequalities: When dividing by a negative number to isolate x, you forget to flip the inequality symbol (> becomes <). Always check the sign of the number you are moving.

3. Variable in Denominator: When solving equations like x + \frac{1}{x} = 2, remember that x cannot be 0. Always check for undefined values.

4. Phantom Negatives: In the quadratic formula, if b is negative (e.g., -5), then -b is positive (5). Students often double-negative incorrectly or drop the sign entirely.

5. Rate vs. Time: In exponential growth y = ab^x, ensure the units of x match the growth rate (e.g., if the rate is monthly, x must be in months, not years).

6. Extraneous Solutions: When you square both sides of a radical equation, you must plug your answers back into the original equation. Often, one answer will result in an impossible statement (like 4 = -4).

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GPT 5.2 Pro

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What You Need to Know

• The ACT Math test is 60 multiple-choice questions in 60 minutes (calculator allowed)—and the official ACT content description lists Algebra as about 12–15 questions, so speed + accuracy on core algebra skills pays off.

• Most Algebra questions test your ability to rewrite expressions, solve equations/inequalities, and model word problems into equations you can solve efficiently.

• Always check for domain restrictions (denominators, radicals) and extraneous solutions (especially in radical equations).

• Modeling shows up throughout ACT Math—practice translating words to equations and interpreting what solutions mean in context.

Linear Equations and Inequalities

Core idea

Linear equation: an equation whose graph is a line, often written as y = mx + b where slope m is the rate of change and intercept b is the value when x = 0.

Linear inequality: like a linear equation but with <, \le, >, \ge; solutions form an interval (or a half-plane in two variables).

Key skills

• Solve for a variable (including distributing and combining like terms).

• Rearrange formulas (literal equations), e.g., solve for x in terms of other variables.

• Solve inequalities—remember multiplying/dividing by a negative flips the sign.

Worked example (equation)

Solve 3(2x - 5) = 4x + 7.
1) Distribute: 6x - 15 = 4x + 7
2) Subtract 4x: 2x - 15 = 7
3) Add 15: 2x = 22
4) Divide by 2: x = 11

Worked example (inequality)

Solve -2x + 3 \le 11.
1) Subtract 3: -2x \le 8
2) Divide by -2 (flip sign): x \ge -4

Exam Focus

• Why it matters: Linear manipulation is the most frequent algebra skill and is often embedded in word problems and modeling.

• Typical question patterns:

  • Solve a one-variable equation with fractions/parentheses.

  • Solve an inequality and choose the correct interval.

  • Rearrange a formula to isolate a variable.

• Common mistakes:

  • Forgetting to flip the inequality when dividing by a negative.

  • Distributing incorrectly, e.g., missing a negative sign.

  • Mixing up slope and intercept when interpreting y = mx + b.

Systems of Equations

Core idea

A system of equations is two (or more) equations solved simultaneously. Solutions represent intersection points.

Types of solutions (two linear equations):

• One solution: lines intersect once.

• No solution: parallel distinct lines.

• Infinitely many: same line (equations are multiples).

Methods

• Substitution: solve one equation for a variable, substitute into the other.

• Elimination: add/subtract equations to cancel a variable.

Worked example (elimination)

Solve
\begin{cases}
2x + y = 11 \
3x - y = 4
\end{cases}
Add equations to eliminate y:
5x = 15 \Rightarrow x = 3
Substitute into 2x + y = 11:
2(3) + y = 11 \Rightarrow y = 5
Solution: (3, 5).

Exam Focus

• Why it matters: Systems test multi-step algebra and appear in coordinate geometry and real-world contexts (mixtures, rates, pricing).

• Typical question patterns:

  • Solve a system and select the ordered pair.

  • Determine number of solutions from equation structure.

  • Word problem leading to two equations.

• Common mistakes:

  • Sign errors when adding/subtracting equations.

  • Substituting into the wrong equation or wrong variable.

  • Reporting x only (forgetting to find y too).

Polynomial Expressions and Equations

Core idea

A polynomial is a sum of terms a_n x^n with nonnegative integer exponents n.

Key vocabulary:

• Degree: highest exponent.

• Factor: an expression that multiplies to make the polynomial.

• Zero/root: a value of x that makes the polynomial equal 0.

High-yield operations

• Combine like terms.

• Multiply polynomials (distributive property/FOIL).

• Factor common factors.

Worked example (multiply)

Expand (x - 4)(x + 7):
x^2 + 7x - 4x - 28 = x^2 + 3x - 28

Worked example (solve by factoring)

Solve x^2 + 3x - 28 = 0.
Factor: x^2 + 3x - 28 = (x + 7)(x - 4)
Set each factor to 0:
x + 7 = 0 \Rightarrow x = -7
x - 4 = 0 \Rightarrow x = 4

Exam Focus

• Why it matters: Polynomial fluency supports quadratics, factoring, and function behavior—common in mid-to-hard ACT questions.

• Typical question patterns:

  • Expand or factor an expression.

  • Solve a polynomial equation by factoring.

  • Use structure (common factor, difference of squares).

• Common mistakes:

  • Dropping terms when distributing.

  • Incorrect sign in the middle term of a product.

  • Factoring errors (especially with negatives).

Quadratic Equations and Factoring

Core idea

A quadratic has the form ax^2 + bx + c with a \ne 0.

Common forms:

• Standard form: ax^2 + bx + c

• Vertex form: a(x - h)^2 + k (vertex at (h, k))

Solving methods

• Factoring (fastest when it works).

• Completing the square (connects to vertex form).

• Quadratic formula:
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

• Discriminant: b^2 - 4ac indicates number of real solutions.

Worked example (quadratic formula)

Solve 2x^2 + x - 3 = 0.
Here a = 2, b = 1, c = -3.
x = \frac{-1 \pm \sqrt{1 - 4(2)(-3)}}{2\cdot 2} = \frac{-1 \pm \sqrt{1 + 24}}{4} = \frac{-1 \pm 5}{4}
So x = 1 or x = -\frac{3}{2}.

Exam Focus

• Why it matters: Quadratics are a cornerstone of ACT algebra—factoring and interpreting roots/vertex are frequent.

• Typical question patterns:

  • Factor and solve a quadratic.

  • Identify vertex/intercepts from an equation.

  • Use the discriminant to determine number of real solutions.

• Common mistakes:

  • Using the quadratic formula with wrong signs (especially -b).

  • Forgetting to set the equation equal to 0 before factoring.

  • Arithmetic errors under the square root.

Radical Expressions and Equations

Core idea

A radical expression involves roots such as \sqrt{\,\,}. For real numbers, remember \sqrt{x} requires x \ge 0.

Simplifying radicals

• Factor out perfect squares: \sqrt{50} = \sqrt{25\cdot 2} = 5\sqrt{2}.

• Rationalize denominators when needed: \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}.

Solving radical equations

Typical process:
1) Isolate the radical.
2) Square both sides.
3) Solve the resulting equation.
4) Check for extraneous solutions in the original.

Worked example (extraneous solution)

Solve \sqrt{x + 5} = x - 1.
Domain: right side must be nonnegative, so x - 1 \ge 0 \Rightarrow x \ge 1.
Square both sides:
x + 5 = (x - 1)^2 = x^2 - 2x + 1
Rearrange:
0 = x^2 - 3x - 4 = (x - 4)(x + 1)
Candidates: x = 4 or x = -1.
Check:

• x = 4: \sqrt{9} = 3 and 4 - 1 = 3 ✓

• x = -1 violates x \ge 1 (and fails original) ✗
  Solution: x = 4.

Exam Focus

• Why it matters: Radicals test algebraic manipulation plus attention to domain—ACT often includes an extraneous-solution trap.

• Typical question patterns:

  • Simplify a radical or rationalize.

  • Solve a radical equation and choose the valid solution.

  • Combine radicals with like terms, e.g., 2\sqrt{5} + 3\sqrt{5}.

• Common mistakes:

  • Not checking solutions after squaring.

  • Incorrectly simplifying, e.g., \sqrt{a+b} \ne \sqrt{a} + \sqrt{b}.

  • Ignoring restrictions like x \ge 0 inside a square root.

Exponential Equations and Growth/Decay

Core idea

An exponential expression has the variable in the exponent, e.g., a\cdot b^x.

Exponent rules (high yield)

• b^m \cdot b^n = b^{m+n}

• \frac{b^m}{b^n} = b^{m-n}

• (b^m)^n = b^{mn}

• b^0 = 1 for b \ne 0

Solving exponential equations (ACT style)

ACT often designs problems solvable by rewriting to the same base.

Worked example:
Solve 3^{2x - 1} = 27.
Rewrite 27 = 3^3.
So 3^{2x - 1} = 3^3 \Rightarrow 2x - 1 = 3 \Rightarrow x = 2.

Growth/decay modeling

A common model is
A(t) = A_0(1 + r)^t
where r is the growth rate per period (negative for decay), and t is number of periods.

Exam Focus

• Why it matters: Exponentials appear in modeling (population, finance) and test equation-solving by structure.

• Typical question patterns:

  • Rewrite both sides with a common base to solve for x.

  • Compare growth rates or compute a future value using A_0(1+r)^t.

  • Interpret what r and A_0 mean in context.

• Common mistakes:

  • Adding exponents incorrectly, e.g., thinking b^{m+n} = b^m + b^n.

  • Confusing linear vs exponential growth (constant difference vs constant ratio).

  • Using percent incorrectly (e.g., 5\% as 5 instead of 0.05).

Graphing and Modeling Expressions

Core idea

Modeling means representing a real situation with an expression or equation, then interpreting the solution.

Linear graphs

• Slope from two points (x1,y1) and (x2,y2):
  m = \frac{y2 - y1}{x2 - x1}

• Point-slope form:
  y - y1 = m(x - x1)

Intercepts and meaning

• x-intercept: set y = 0.

• y-intercept: set x = 0.
  In word problems, interpret intercepts carefully (they may not be realistic if negative or non-integer).

Quadratic graphs (key features)

For y = ax^2 + bx + c:

• Axis of symmetry: x = -\frac{b}{2a}

• Vertex occurs at that x value.

• Roots are where y = 0.

Worked modeling example (translate words)

A gym charges a \$25 sign-up fee plus \$15 per month. Write a cost model after m months.
Let C be total cost:
C = 25 + 15m
Interpretation: 25 is the initial value (intercept), 15 is the rate (slope).

Exam Focus

• Why it matters: The ACT explicitly emphasizes modeling across topics; many “algebra” items are really translation + interpretation.

• Typical question patterns:

  • Build an equation from a scenario (rates, costs, mixture, perimeter/area with algebra).

  • Identify slope/intercepts from a graph description or equation.

  • Match an equation to a described transformation or feature (vertex, intercepts).

• Common mistakes:

  • Mixing up what the variable represents (months vs total cost).

  • Using the wrong rate (confusing starting value with change per unit).

  • Solving correctly but choosing an answer that ignores context (negative time, impossible counts).

Quick Review Checklist

• Can you solve linear equations with parentheses and fractions accurately?

• Can you solve linear inequalities and remember when to flip the inequality sign?

• Can you solve a system using substitution or elimination and interpret what the solution means?

• Can you expand and factor polynomials (including common factors and products of binomials)?

• Can you solve quadratics by factoring and by the quadratic formula?

• Do you know how to simplify radicals and rationalize denominators?

• Can you solve a radical equation and check for extraneous solutions?

• Can you solve exponential equations by rewriting to a common base?

• Can you build a linear or exponential model from a word problem and interpret slope/intercept or growth rate?

Final Exam Pitfalls

1. Not checking domain/extraneous solutions—after squaring a radical equation, always plug solutions back into the original.

2. Inequality sign errors—when you multiply/divide by a negative, flip < to > (and \le to \ge).

3. Forgetting to set quadratics equal to zero—factoring methods require ax^2+bx+c=0 before using the zero-product property.

4. Exponent rule mix-ups—remember b^{m+n} comes from multiplying same-base powers, not adding them.

5. Modeling misreads—define variables first, track units (dollars, months, miles), and reject answers that don’t make real-world sense.

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Claude Opus 4.6

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What You Need to Know

• Algebra accounts for a significant portion of the ACT Math section — roughly 12–15 of the 60 questions fall under the "Preparing for Higher Math: Algebra" category, making it one of the highest-weighted content domains.

• You must be fluent in manipulating expressions and solving equations across multiple forms: linear, quadratic, polynomial, radical, and exponential. Speed and accuracy with these fundamentals will earn you the most points.

• The ACT tests algebra in context — expect word problems that require you to set up an equation or inequality before solving, and questions that ask you to interpret graphs or models rather than just compute.

• Many algebra questions are designed with trap answer choices that correspond to common sign errors, distribution mistakes, or misreading the question. Careful, systematic work beats rushing every time.

Linear Equations and Inequalities

A linear equation is any equation that can be written in the form ax + b = c, where the variable has an exponent of 1. A linear inequality replaces the equals sign with <, >, \leq, or \geq.

Key Skills

• Solve one-variable equations by isolating x: use inverse operations, combine like terms, distribute as needed.

• Solve and graph one-variable inequalities. Remember: when you multiply or divide both sides by a negative number, flip the inequality sign.

• Translate word problems into linear equations or inequalities.

Slope-Intercept and Standard Forms

Slope measures rate of change: m = \frac{y2 - y1}{x2 - x1}.

Example

Solve 3(x - 4) + 2 = 5x - 6:

1. Distribute: 3x - 12 + 2 = 5x - 6

2. Combine: 3x - 10 = 5x - 6

3. Subtract 3x: -10 = 2x - 6

4. Add 6: -4 = 2x

5. Divide: x = -2

Exam Focus

• Why it matters: Linear equations and inequalities appear on nearly every ACT. They range from easy to medium difficulty and are reliable point-earners.

• Typical question patterns:

  • Solve for x in a multi-step equation.

  • "Which inequality represents the situation described?"

  • Find the slope or y-intercept from a table, graph, or two points.

• Common mistakes:

  • Forgetting to flip the inequality sign when dividing by a negative.

  • Sign errors when distributing a negative across parentheses.

  • Misidentifying slope vs. y-intercept in y = mx + b.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. On the ACT, you'll almost always see two-variable, two-equation systems.

Methods of Solving

• Substitution: Solve one equation for a variable, substitute into the other.

• Elimination (Addition): Multiply equations so that one variable cancels when you add or subtract the equations.

• Graphing interpretation: The solution is the point of intersection.

Example (Elimination)

Solve: 2x + 3y = 12 and 4x - 3y = 6

1. Add the equations: 6x = 18

2. x = 3

3. Substitute back: 2(3) + 3y = 12 \Rightarrow 3y = 6 \Rightarrow y = 2

4. Solution: (3, 2)

No solution means the lines are parallel (same slope, different intercepts). Infinitely many solutions means the equations represent the same line.

Exam Focus

• Why it matters: Systems questions appear regularly and can be solved quickly with the right method — often faster by elimination than substitution.

• Typical question patterns:

  • Solve for one specific variable (not both).

  • Word problems: "Two items cost… find the price of each."

  • Determine the number of solutions a system has.

• Common mistakes:

  • Solving for x when the question asks for y (or vice versa).

  • Arithmetic errors when multiplying an entire equation by a constant before elimination.

  • Not checking that the answer satisfies both equations.

Polynomial Expressions and Equations

A polynomial is an expression with one or more terms, each consisting of a coefficient and a variable raised to a non-negative integer power. Examples: 3x^2 + 5x - 7, x^3 - 1.

Key Operations

• Adding/Subtracting: Combine like terms.

• Multiplying: Use the distributive property (FOIL for binomials). For (a + b)(c + d) = ac + ad + bc + bd.

• Factoring: Pull out the greatest common factor (GCF) first, then look for patterns.

Special Products to Memorize

Exam Focus

• Why it matters: Polynomial manipulation is the foundation for harder algebra and function questions. Expect 2–4 questions directly testing these skills.

• Typical question patterns:

  • Simplify or expand a polynomial expression.

  • Factor a polynomial completely.

  • Determine the degree of a polynomial or identify equivalent expressions.

• Common mistakes:

  • Forgetting to distribute the negative sign when subtracting polynomials: -(3x - 2) = -3x + 2, not -3x - 2.

  • Mixing up the difference of squares with a sum of squares (a^2 + b^2 does not factor over the reals).

Quadratic Equations and Factoring

A quadratic equation has the standard form ax^2 + bx + c = 0. The graph of a quadratic function is a parabola.

Methods of Solving

1. Factoring: Express as (x - r)(x - s) = 0, then x = r or x = s.

2. Quadratic Formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

3. Completing the square: Rewrite in vertex form a(x - h)^2 + k.

The discriminant D = b^2 - 4ac tells you the number of real solutions:

• D > 0: two distinct real solutions

• D = 0: one repeated real solution

• D < 0: no real solutions

Example

Solve x^2 - 5x + 6 = 0 by factoring:

1. Find two numbers that multiply to 6 and add to -5: -2 and -3.

2. (x - 2)(x - 3) = 0

3. x = 2 or x = 3

Exam Focus

• Why it matters: Quadratics are among the most frequently tested topics on the ACT Math section — expect 3–5 questions.

• Typical question patterns:

  • Solve a quadratic equation (factoring or formula).

  • Find the vertex, axis of symmetry, or roots from a given equation.

  • Use the discriminant to determine the number of solutions.

• Common mistakes:

  • Forgetting the \pm in the quadratic formula and finding only one root.

  • Setting up the formula with the wrong sign for b (remember it's -b).

  • Not setting the equation equal to zero before factoring or applying the formula.

Radical Expressions and Equations

A radical expression contains a root symbol, most commonly a square root: \sqrt{x}. A radical equation has the variable inside the radical.

Solving Radical Equations

1. Isolate the radical on one side.

2. Square both sides (or raise to the appropriate power) to eliminate the radical.

3. Solve the resulting equation.

4. Check for extraneous solutions — squaring can introduce answers that don't satisfy the original equation.

Simplification Rules

• \sqrt{ab} = \sqrt{a} \cdot \sqrt{b}

• \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}

• \sqrt{x^2} = |x|

Example

Solve \sqrt{x + 3} = 5:

1. Square both sides: x + 3 = 25

2. x = 22

3. Check: \sqrt{22 + 3} = \sqrt{25} = 5 ✓

Exam Focus

• Why it matters: Radical questions appear less frequently (1–2 questions) but are straightforward points if you know the technique.

• Typical question patterns:

  • Solve a radical equation for x.

  • Simplify a radical expression.

  • Determine the domain of a radical function.

• Common mistakes:

  • Forgetting to check for extraneous solutions after squaring.

  • Assuming \sqrt{a + b} = \sqrt{a} + \sqrt{b} — this is false.

  • Not isolating the radical before squaring.

Exponential Equations and Growth/Decay

Exponential functions have the form y = a \cdot b^x, where a is the initial value and b is the base (growth/decay factor).

• If b > 1: exponential growth

• If 0 < b < 1: exponential decay

Growth/Decay Model

y = a(1 + r)^t for growth, y = a(1 - r)^t for decay, where r is the rate and t is time.

Key Exponent Rules

Example

A population of 500 bacteria doubles every 3 hours. Write an equation and find the population after 9 hours.

• y = 500 \cdot 2^{t/3}

• At t = 9: y = 500 \cdot 2^{9/3} = 500 \cdot 2^3 = 500 \cdot 8 = 4000

Exam Focus

• Why it matters: Exponential questions test both algebraic manipulation (exponent rules) and real-world modeling. Expect 1–3 questions.

• Typical question patterns:

  • Simplify expressions using exponent rules.

  • Identify growth vs. decay from an equation or context.

  • Predict a future value given an exponential model.

• Common mistakes:

  • Confusing a^m \cdot a^n = a^{m+n} with a^m \cdot b^m = (ab)^m — mixing up which rule applies.

  • Adding bases instead of multiplying: 2^3 \cdot 2^4 \neq 2^{12} (it equals 2^7).

  • Misreading the growth rate — a 5% increase means b = 1.05, not b = 0.05.

Graphing and Modeling Expressions

This topic asks you to connect algebraic expressions and equations to their graphical representations and to real-world contexts.

Core Concepts

• Linear models produce straight lines; look for constant rates of change.

• Quadratic models produce parabolas; look for a maximum/minimum and symmetry.

• Exponential models produce curves that grow or decay rapidly; look for percentage-based change.

• The y-intercept represents the initial value in a model (when x = 0 or t = 0).

• The slope (linear) or growth factor (exponential) represents the rate of change.

Interpreting Graphs

When given a graph, identify:

• Intercepts (where the graph crosses the axes)

• Slope or rate of change

• Whether the function is increasing, decreasing, or constant

• Key features like vertex (quadratic), asymptotes (exponential)

Choosing the Right Model

Exam Focus

• Why it matters: The ACT increasingly emphasizes interpreting models and graphs — these questions test conceptual understanding, not just computation.

• Typical question patterns:

  • "Which equation best models the data shown in the table/graph?"

  • "What does the slope (or y-intercept) represent in context?"

  • Match a given equation to its graph.

• Common mistakes:

  • Confusing linear and exponential growth — linear adds a constant, exponential multiplies by a constant.

  • Misreading graph scales or axis labels.

  • Choosing a model based on appearance without checking whether data points fit.

Quick Review Checklist

• Can you solve a multi-step linear equation and inequality, including remembering to flip the inequality sign?

• Can you solve a system of two equations using both substitution and elimination?

• Can you multiply and factor polynomials, including recognizing difference of squares and perfect square trinomials?

• Can you solve a quadratic equation by factoring, and can you apply the quadratic formula correctly?

• Do you know how to use the discriminant to determine the number of real solutions?

• Can you solve a radical equation and check for extraneous solutions?

• Do you know all the key exponent rules (product, quotient, power, zero, negative)?

• Can you set up and interpret an exponential growth/decay model?

• Can you identify whether data is best modeled by a linear, quadratic, or exponential function?

• Can you interpret slope, y-intercept, and vertex in the context of a word problem?

Final Exam Pitfalls

1. Solving for the wrong variable: The ACT loves asking for y when your instinct is to solve for x, or asking for the value of an expression like 3x + 2 instead of just x. Always re-read the question before selecting your answer.

2. Sign errors with negatives: Distributing -(x - 5) as -x - 5 instead of -x + 5 is one of the most common errors. Slow down when negatives are involved.

3. Forgetting to check extraneous solutions: After squaring both sides of a radical equation, one or both solutions may not work in the original equation. Always plug answers back in.

4. Not setting quadratics equal to zero: You cannot factor or use the quadratic formula until the equation is in the form ax^2 + bx + c = 0. If the equation reads x^2 + 3x = 10, subtract 10 first.

5. Confusing growth rate with growth factor: A 7% annual growth rate means the growth factor is 1.07, not 0.07. The decay version of 7% means the factor is 0.93.

6. Misapplying exponent rules: 2^3 + 2^4 \neq 2^7. Exponent rules apply to multiplication and division of like bases, not addition and subtraction.