SAT Math: Concepts, Methods, and Worked Problems

Linear Relationships and Equations

Linear relationships are the backbone of SAT Math because they model “constant rate of change” situations: every time x increases by 1, y changes by the same amount. Many real contexts behave approximately linearly over a range—wages per hour, cost per item, distance at constant speed—so the SAT uses lines to test both algebra skills and modeling.

What a linear equation means (slope and intercept)

A linear equation in two variables describes a straight line when graphed in the coordinate plane. The most common form is slope-intercept form:

y = mx + b

Here, m is the slope, meaning the change in y for a 1-unit increase in x. The number b is the y-intercept, the value of y when x = 0. If you remember only one interpretation, remember this: slope is “steepness” (rate), intercept is “starting value.”

A second extremely useful form is point-slope form:

y - y1 = m(x - x1)

This is the same idea, but it highlights a known point (x1, y1) on the line and the slope m.

A common misconception is thinking slope is “rise over run” only from left to right. It is, but the key is consistency: you must use the same direction for rise and run. For example, moving right by 2 (run = 2) and up by 6 (rise = 6) gives slope 6/2 = 3. Moving left by 2 and down by 6 also gives slope (-6)/(-2)=3.

Finding slope from two points

When you’re given two points, slope comes from the ratio of vertical change to horizontal change:

m = \frac{y2 - y1}{x2 - x1}

The order doesn’t matter as long as you keep it consistent in numerator and denominator. A classic SAT error is mixing orders (like y2 - y1 over x1 - x2), which flips the sign.

Example (slope and equation): Find the equation of the line through (2, -1) and (6, 7) .

1. Compute slope:

m = \frac{7 - (-1)}{6 - 2} = \frac{8}{4} = 2

2. Use point-slope form with (2, -1) :

y - (-1) = 2(x - 2)

3. Simplify:

y + 1 = 2x - 4

y = 2x - 5

So the line is y = 2x - 5.

Parallel and perpendicular lines

Slope also controls angle relationships between lines.

• Parallel lines have the same slope.
• Perpendicular lines have slopes that are negative reciprocals (as long as both are non-vertical).

If one line has slope m, a perpendicular line has slope:

m_\perp = -\frac{1}{m}

Vertical lines have equations x = c (undefined slope). Horizontal lines have equations y = c (slope 0). Vertical and horizontal lines are perpendicular.

Example (perpendicular slope): A line has slope -3/5. A perpendicular line has slope:

m_\perp = -\frac{1}{-3/5} = \frac{5}{3}

Solving linear equations efficiently

Solving linear equations is about keeping expressions equivalent by performing the same operation to both sides.

A useful mindset: your goal is to “isolate the variable,” but you should also watch for opportunities to simplify early (distribute, combine like terms) to avoid mistakes.

Example (multi-step): Solve for x:

3(2x - 5) + 4 = 2x + 19

Distribute and combine:

6x - 15 + 4 = 2x + 19

6x - 11 = 2x + 19

Subtract 2x:

4x - 11 = 19

Add 11:

4x = 30

Divide by 4:

x = \frac{15}{2}

A common error here is dropping a negative during distribution or when moving terms across the equals sign.

Linear inequalities and solution sets

Linear inequalities look like equations but represent ranges of values.

Example forms:

2x + 3 \le 11

y > -x + 4

Two key SAT rules:

1. You solve inequalities like equations.
2. If you multiply or divide by a negative, you must flip the inequality sign.

Example (inequality): Solve:

-4x + 1 \ge 9

Subtract 1:

-4x \ge 8

Divide by -4 and flip the sign:

x \le -2

On a number line, that’s all values at or below -2.

Systems of linear equations (meaning and methods)

A system of equations asks for values that satisfy two equations at the same time. Graphically, you’re looking for where two lines intersect.

You typically see three cases:

• One solution (lines intersect once)
• No solution (parallel distinct lines)
• Infinitely many solutions (same line written differently)

Two main algebra methods:

1. Substitution: solve one equation for a variable, plug into the other.
2. Elimination: add/subtract equations to cancel a variable.

Example (elimination):

2x + 3y = 13

4x - 3y = 5

Add the equations to eliminate y:

6x = 18

x = 3

Substitute into the first equation:

2(3) + 3y = 13

6 + 3y = 13

3y = 7

y = \frac{7}{3}

So the solution is \left(3, \frac{7}{3}\right).

Modeling with linear equations

SAT modeling often gives you words, a table, or a graph and asks you to create or interpret an equation.

• Slope m becomes a rate (per unit of x).
• Intercept b becomes an initial value (when x = 0).

Be careful: sometimes x = 0 is not meaningful in context, but the intercept still exists algebraically.

Example (context): A taxi charges a base fee of 4 dollars plus 2 dollars per mile. Let m be miles and C be total cost.

The starting cost at m = 0 is 4, so intercept is 4. The rate per mile is 2, so slope is 2:

C = 2m + 4

Exam Focus

• Typical question patterns:
  • Given two points or a graph, find slope, intercept, or an equation of the line.
  • Solve a system and interpret what the solution means in context.
  • Write or interpret a linear model from a table or word problem.
• Common mistakes:
  • Sign errors in slope (swapping order in numerator but not denominator).
  • Forgetting to flip the inequality when dividing by a negative.
  • Treating parallel lines as if they intersect, or missing the “no solution” case.

Functions and Nonlinear Relationships

A function is a rule that assigns each input exactly one output. SAT questions use functions to test your ability to connect algebraic expressions, tables, and graphs—and to interpret what inputs and outputs mean in context.

Function notation and interpretation

Function notation looks like this:

f(x)

Read it as “the output of function f when the input is x.” The biggest conceptual trap is thinking f(x) means f \times x. It does not. It is a single value.

If a problem says f(3) = 10, that means input 3 produces output 10.

The domain is the set of allowed inputs. The range is the set of resulting outputs. On SAT, domain is often restricted by:

• denominators (cannot be zero)
• even roots (radicand must be nonnegative for real outputs)
• context (time can’t be negative, number of people is an integer, etc.)

Working with function expressions

Common tasks include evaluation, solving equations involving functions, and interpreting changes.

Example (evaluate): If

f(x) = 2x^2 - 3x + 1

then

f(-2) = 2(-2)^2 - 3(-2) + 1 = 2(4) + 6 + 1 = 15

When substituting negative numbers, parentheses are essential to avoid sign mistakes.

Graph features: intercepts, vertex, maxima/minima

Graphs encode key information quickly.

• An x-intercept occurs where y = 0.
• A y-intercept occurs where x = 0.
• For many nonlinear functions (especially quadratics), turning points matter.

Quadratic functions (parabolas)

A quadratic function has the form:

f(x) = ax^2 + bx + c

Its graph is a parabola. If a > 0 it opens upward (minimum). If a < 0 it opens downward (maximum).

A particularly meaningful form is vertex form:

f(x) = a(x - h)^2 + k

The point (h, k) is the vertex (the turning point). This form is great for identifying maximum/minimum values.

You can also find the vertex of ax^2 + bx + c using:

x = -\frac{b}{2a}

Then plug that x into the function to get y.

Solving quadratics (factoring, quadratic formula)

Solving ax^2 + bx + c = 0 means finding the x-values where the parabola crosses the x-axis.

Factoring works when the quadratic factors nicely into:

(px + r)(qx + s) = 0

Then each factor gives a solution.

When factoring is hard or impossible with integers, use the quadratic formula:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the square root is the discriminant:

D = b^2 - 4ac

• If D > 0: two real solutions
• If D = 0: one real solution (double root)
• If D < 0: no real solutions (complex solutions)

Example (quadratic formula): Solve

2x^2 + 3x - 2 = 0

Here a = 2, b = 3, c = -2.

x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}

x = \frac{-3 \pm \sqrt{9 + 16}}{4}

x = \frac{-3 \pm 5}{4}

So

x = \frac{1}{2}

or

x = -2

A frequent mistake is substituting c without its sign.

Polynomial expressions and factoring structure

SAT algebra often tests whether you recognize structure, such as difference of squares:

a^2 - b^2 = (a - b)(a + b)

and perfect square trinomials:

x^2 + 2xy + y^2 = (x + y)^2

x^2 - 2xy + y^2 = (x - y)^2

Factoring matters because it reveals zeros (solutions), simplifies rational expressions, and helps you solve equations faster.

Example (difference of squares):

x^2 - 49 = 0

Factor:

(x - 7)(x + 7) = 0

So x = 7 or x = -7.

Rational expressions (fractions with variables)

A rational expression is a fraction of polynomials, like:

\frac{x^2 - 1}{x - 1}

Two must-know ideas:

1. You can simplify by factoring and canceling common factors.
2. You can never allow the original denominator to be zero.

For the example:

\frac{x^2 - 1}{x - 1} = \frac{(x - 1)(x + 1)}{x - 1} = x + 1

But the original expression is undefined at x = 1, so the simplified form represents the same values except it must keep the restriction x \ne 1.

Exponents and radicals (growth and constraints)

Exponent rules show up constantly. The SAT expects you to use them accurately, not memorize dozens of tricks.

Key rules (for real numbers where defined):

a^m a^n = a^{m+n}

\frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn}

Also:

a^{1/2} = \sqrt{a}

Radicals introduce domain restrictions: \sqrt{x} requires x \ge 0 if you’re staying in real numbers.

Example (simplify):

\frac{(x^3 y^2)^2}{x^4 y}

Compute the power:

(x^3 y^2)^2 = x^6 y^4

Divide:

\frac{x^6 y^4}{x^4 y} = x^2 y^3

A common error is adding exponents when dividing (it should be subtraction).

Exponential functions (repeated multiplication)

Exponential models describe repeated percent change: growth or decay.

A basic exponential form is:

y = a b^x

• a is the initial value when x = 0.
• b is the growth factor per 1 unit of x.

If something grows by 5% each period, then b = 1.05. If it decreases by 12% each period, then b = 0.88.

Example (percent growth): A population is 2000 and grows 3% per year. After t years:

P = 2000(1.03)^t

A frequent mistake is using 0.03 instead of 1.03.

Exam Focus

• Typical question patterns:
  • Evaluate functions, interpret f(a) - f(b), or solve f(x) = k.
  • Quadratics: find vertex, zeros, or rewrite between standard and vertex form.
  • Simplify expressions with exponents/radicals or rational expressions with domain restrictions.
• Common mistakes:
  • Treating f(x) like multiplication and losing the idea of “input-output.”
  • Canceling terms across addition (you can cancel factors, not separate terms).
  • Forgetting domain restrictions (like denominator not zero or radicand nonnegative).

Problem Solving with Ratios, Rates, and Percent

A large part of SAT Math is translating everyday quantitative language into equations. Ratios, unit rates, and percent change are the “grammar” of applied math problems—if you can express the relationships correctly, the algebra usually becomes straightforward.

Ratios and proportional relationships

A ratio compares quantities by division. If the ratio of A to B is 3:5, that means:

\frac{A}{B} = \frac{3}{5}

A relationship is proportional if one quantity equals a constant times the other:

y = kx

The constant k is the constant of proportionality. Graphically, proportional relationships are lines through the origin.

Example (proportion): If 8 notebooks cost 12 dollars at a constant rate, the unit rate is:

\frac{12}{8} = 1.5

So cost C for n notebooks is:

C = 1.5n

A common mistake is flipping the rate (writing notebooks per dollar when you need dollars per notebook).

Rates and unit analysis (dimensional reasoning)

A rate is a ratio with units, like miles per hour. SAT problems often become easy if you track units like algebra.

If you travel at 60 miles per hour for 2.5 hours, distance is:

d = rt

where r is rate and t is time.

d = 60 \cdot 2.5 = 150

Unit analysis also prevents errors in conversions.

Example (conversion): If a car travels 90 miles in 1.5 hours, speed is:

\frac{90}{1.5} = 60

So 60 miles per hour.

Percent, percent change, and reverse percent

A percent means “per 100.” So 35% is:

0.35

The formula for percent change from original A to new B is:

\text{percent change} = \frac{B - A}{A} \times 100\%

But on SAT, it’s often faster to use a multiplier:

• Increase by p percent: multiply by 1 + p (with p as a decimal).
• Decrease by p percent: multiply by 1 - p.

Example (discount): An item costs 80 dollars and is discounted by 25%. New price:

80(1 - 0.25) = 80(0.75) = 60

Reverse percent (finding the original)

If you’re told the final value after a percent change and asked for the original, divide by the multiplier.

Example (original price): After a 20% discount, a price is 48 dollars. Original price P satisfies:

0.8P = 48

P = \frac{48}{0.8} = 60

A common error is subtracting 20% of 48 instead of undoing the multiplier.

Mixtures and weighted averages

Mixture problems are about combining quantities with different “concentrations” (or values) to produce a combined concentration.

If you mix a units at value v1 with b units at value v2, the average value is:

\frac{av1 + bv2}{a + b}

Example (weighted average): A student scores 80 on a test worth 40% of the grade and 95 on a test worth 60%. Overall grade:

0.4(80) + 0.6(95) = 32 + 57 = 89

A common mistake is averaging 80 and 95 directly without weights.

Direct and inverse variation

Variation language is a frequent SAT translation skill.

• Direct variation: y varies directly with x means:

y = kx

• Inverse variation: y varies inversely with x means:

y = \frac{k}{x}

Example (inverse variation): If y varies inversely with x and y = 6 when x = 2, then:

6 = \frac{k}{2}

k = 12

So:

y = \frac{12}{x}

A typical mistake is writing y = kx when the problem says “inversely.”

Exam Focus

• Typical question patterns:
  • Translate a word description into a proportion, a linear model, or a variation equation.
  • Percent increase/decrease, including successive percent changes.
  • Weighted average or mixture contexts (solutions, prices, grades).
• Common mistakes:
  • Using the wrong base for percent change (the denominator should be the original).
  • Confusing “percent of” with “percent increase.”
  • Averaging values without accounting for weights or quantities.

Data Analysis, Statistics, and Probability

SAT data questions test whether you can read and reason from information—not just compute. You’re expected to understand what summary statistics mean, compare groups fairly, and interpret probabilities from tables or studies.

Reading tables, graphs, and units

Before calculating anything, identify:

• What each axis or column represents
• The units (minutes vs hours, thousands vs ones)
• Whether values are totals, rates, or averages

Many SAT “trick” errors come from ignoring scale labels (like “in thousands”) or mixing units.

Measures of center: mean and median

The mean is the arithmetic average:

\text{mean} = \frac{\text{sum of values}}{\text{number of values}}

The median is the middle value when data are ordered (or the average of the two middle values if there are an even number of data points).

Why both matter: the mean uses every value and is sensitive to outliers; the median is resistant to extreme values.

Example (outlier effect): Data: 2, 3, 3, 4, 30.

Mean:

\frac{2+3+3+4+30}{5} = \frac{42}{5} = 8.4

Median is 3 (middle of ordered list). The mean is pulled upward by 30; the median reflects the typical small values.

Measures of spread: range and standard deviation (conceptual)

The range is:

\text{range} = \text{max} - \text{min}

SAT also references standard deviation conceptually: a measure of typical distance from the mean. You may be asked which of two data sets has greater standard deviation based on how spread out the values are, not to compute the full formula.

A key idea: if one set has the same mean but values are more dispersed, it has larger standard deviation.

Scatterplots and lines of best fit

A scatterplot shows pairs (x, y) . If the pattern is roughly linear, a line of best fit approximates the trend.

You may be asked:

• Is association positive, negative, or none?
• Use the line to predict (interpolate) a value.
• Identify an outlier or interpret residuals.

A residual is:

\text{residual} = \text{actual} - \text{predicted}

If residual is positive, the point lies above the prediction line; if negative, below.

Example (residual): A line predicts y = 50 at x = 10, but actual is y = 56.

\text{residual} = 56 - 50 = 6

Two-way tables and conditional probability

A two-way table counts outcomes for two categories (like “plays a sport” vs “does not,” and “9th grade” vs “10th grade”). Probability questions often require careful attention to the “given” condition.

If A and B are events, then conditional probability is:

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

In table terms: “probability of A given B” equals the count in the A and B cell divided by the total count in the B group.

Example (conditional probability from counts): In a school, 40 students play an instrument; 10 of those also play a sport. If you select a student who plays an instrument, probability they also play a sport is:

\frac{10}{40} = \frac{1}{4}

A classic mistake is dividing by the total number of students instead of the “given” group.

Interpreting study design (association vs causation)

SAT sometimes asks what conclusions can be drawn from a survey or experiment.

• A random sample supports generalizing results to a population.
• A randomized experiment supports cause-and-effect conclusions.

If a study is observational (no random assignment to treatments), you can usually discuss association but not causation.

Exam Focus

• Typical question patterns:
  • Compute or compare mean/median; describe effect of adding/removing a data point.
  • Interpret a scatterplot trend, choose a reasonable prediction, or compute a residual.
  • Conditional probability from a two-way table; interpret “given that.”
• Common mistakes:
  • Mixing up which group is the denominator in conditional probability.
  • Treating correlation as proof of causation when the design doesn’t justify it.
  • Ignoring axis scaling or units and reporting a value off by a factor of 10 or 1000.

Geometry and Trigonometry Essentials

SAT geometry emphasizes core relationships you can apply flexibly: angle facts, triangle similarity, circles, coordinate geometry, and right-triangle trigonometry. The test is less about memorizing obscure theorems and more about connecting a diagram to a small set of reliable tools.

Angle relationships and lines

Key angle facts:

• A straight line is 180 degrees; a full rotation is 360 degrees.
• Vertical angles are equal.
• Adjacent angles on a line sum to 180 degrees.
• When parallel lines are cut by a transversal, corresponding angles are equal and same-side interior angles sum to 180 degrees.

Errors often come from assuming lines are parallel just because they “look” parallel. On SAT, you can only use parallel-line relationships if the problem states or marks them.

Triangles: sum of angles and area

In any triangle, the angle sum is 180 degrees.

Area of a triangle:

A = \frac{1}{2}bh

where b is a base and h is the perpendicular height to that base.

Example (area): If b = 10 and h = 7, then:

A = \frac{1}{2}(10)(7) = 35

Pythagorean theorem and distance in the coordinate plane

For a right triangle with legs a and b and hypotenuse c:

a^2 + b^2 = c^2

This powers the distance formula. The distance between points (x1, y1) and (x2, y2) is:

d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2}

Example (distance): Distance between (1, 2) and (5, 5) :

d = \sqrt{(5-1)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = 5

A common mistake is forgetting to square the differences before adding, especially when differences are negative.

Similar triangles and scaling

Similar triangles have equal corresponding angles and proportional corresponding side lengths. Similarity is one of the most powerful SAT geometry ideas because it turns unknown lengths into proportions.

If triangles are similar with scale factor k, then:

• side lengths multiply by k
• perimeters multiply by k
• areas multiply by k^2

Example (scale factor): A triangle with area 12 is scaled by factor 3. New area:

12 \cdot 3^2 = 12 \cdot 9 = 108

Students often (incorrectly) multiply area by 3 instead of 9.

Circles: radius, diameter, circumference, area

For a circle with radius r:

C = 2\pi r

A = \pi r^2

Diameter d is:

d = 2r

SAT circle problems also use arc length and sector area when angles are given (often in degrees). If a central angle is a fraction of a full circle, that same fraction applies to circumference and area.

If the central angle is \theta degrees, then arc length is:

\text{arc length} = \frac{\theta}{360} \cdot 2\pi r

Sector area is:

\text{sector area} = \frac{\theta}{360} \cdot \pi r^2

Example (arc length): Radius 6, central angle 120 degrees:

\text{arc length} = \frac{120}{360} \cdot 2\pi(6) = \frac{1}{3} \cdot 12\pi = 4\pi

A common mistake is using \theta/180 instead of \theta/360 for degree-based circle fractions.

Coordinate geometry: slope, midpoint, and circle equations

Midpoint of segment from (x1, y1) to (x2, y2) :

\left(\frac{x1+x2}{2}, \frac{y1+y2}{2}\right)

Circle equation (center-radius form) with center (h, k) and radius r:

(x - h)^2 + (y - k)^2 = r^2

You may be asked to identify center and radius from the equation, or build the equation from given information.

Volume and area of 3D figures

The SAT focuses on common solids.

Rectangular prism volume:

V = lwh

Cylinder volume:

V = \pi r^2 h

Right circular cone volume:

V = \frac{1}{3}\pi r^2 h

Sphere volume:

V = \frac{4}{3}\pi r^3

A frequent error is confusing formulas for cone vs cylinder (cone is one-third of the cylinder with same base and height).

Right-triangle trigonometry

Trigonometry on the SAT is mostly right-triangle based.

For an acute angle \theta in a right triangle:

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

A memory aid many students like is “SOH-CAH-TOA” (Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent).

Example (using sine): If \sin(\theta) = 3/5, then opposite:hypotenuse is 3:5. A matching right triangle is 3-4-5, so adjacent is 4 (up to scaling). Then:

\cos(\theta) = \frac{4}{5}

Common mistakes include swapping adjacent and opposite (adjacent depends on which angle you’re referencing) and forgetting that these definitions only apply directly to right triangles.

Exam Focus

• Typical question patterns:
  • Use triangle facts (angle sum, Pythagorean theorem) to find missing sides/angles.
  • Apply similarity to compute unknown lengths or compare areas.
  • Circles: circumference/area, arc length/sector area as fractions of a circle.
• Common mistakes:
  • Using a theorem not justified by markings (assuming right angles or parallel lines).
  • Scaling area/volume linearly instead of with squares/cubes.
  • Mixing up opposite/adjacent in trig when the reference angle changes.

Complex Numbers and Algebraic Structure

Complex numbers appear on the SAT primarily through quadratic equations that don’t have real solutions and through operations like simplifying expressions with i. The goal is not advanced complex analysis; it’s careful algebra and comfort with the definition of i.

What complex numbers are

The imaginary unit i is defined by:

i^2 = -1

A complex number has the form:

a + bi

where a and b are real numbers. If b = 0 the number is real; if a = 0 it’s purely imaginary.

The key simplification pattern is powers of i:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

Then it repeats every 4 powers.

Adding and multiplying complex numbers

Add by combining real parts and imaginary parts.

Example (addition):

(3 + 2i) + (5 - 7i) = 8 - 5i

Multiply using distribution (FOIL) and then use i^2 = -1.

Example (multiplication):

(2 + 3i)(4 - i)

Distribute:

= 8 - 2i + 12i - 3i^2

Combine like terms and replace i^2:

= 8 + 10i - 3(-1)

= 11 + 10i

A common error is leaving i^2 in the final answer or turning i^2 into 1 instead of -1.

Complex solutions from the discriminant

When you solve a quadratic using the quadratic formula, a negative discriminant means you’ll take the square root of a negative number.

If:

D = b^2 - 4ac < 0

then:

\sqrt{D} = \sqrt{-1}\sqrt{|D|} = i\sqrt{|D|}

Example (complex roots): Solve:

x^2 + 4x + 13 = 0

Compute discriminant:

D = 4^2 - 4(1)(13) = 16 - 52 = -36

Quadratic formula:

x = \frac{-4 \pm \sqrt{-36}}{2}

x = \frac{-4 \pm 6i}{2}

So:

x = -2 + 3i

or

x = -2 - 3i

Notice the solutions come as conjugate pairs when coefficients are real.

Equivalent forms and algebraic “structure” moves

SAT often rewards seeing structure rather than expanding everything.

Two key habits:

1. Factor before you expand when solving or simplifying.
2. Isolate a repeated expression by substitution.

Example (substitution structure): Solve for x:

\sqrt{x+5} = 4

Square both sides:

x + 5 = 16

x = 11

But here’s the structural caution: squaring both sides can introduce extraneous solutions in some equations. Always check in the original.

Check:

\sqrt{11+5} = \sqrt{16} = 4

So x=11 works.

A common mistake is forgetting the domain restriction: x + 5 must be nonnegative.

Exam Focus

• Typical question patterns:
  • Simplify expressions involving i and reduce powers of i.
  • Use the quadratic formula and interpret negative discriminants.
  • Solve radical equations and check solutions for validity.
• Common mistakes:
  • Replacing i^2 with 1 instead of -1.
  • Dropping the \pm in the quadratic formula.
  • Solving a radical equation but not checking for extraneous solutions.