Math Formulas/Equations
What You Need to Know
On SAT Math, a huge share of points comes from recognizing which equation/formula models the situation and then manipulating it cleanly. This topic is about your core toolbox:
- Algebra rules (distributing, factoring, exponents/radicals)
- Solving equations/inequalities (linear, quadratic, rational, absolute value)
- Core function models (linear, exponential)
- Coordinate geometry equations (line formulas, circle equation)
If you can (1) rewrite expressions correctly and (2) solve for the variable without illegal moves, you’ll crush most “equation” questions.
Critical reminder: Whatever you do to one side of an equation/inequality, do it to the other side too. The only time the “same operation” rule changes is inequalities: multiplying/dividing by a negative flips the sign.
Core idea
An equation states two expressions are equal and you’re finding values that make it true. An inequality compares expressions with , \ge and you’re finding values that make it true.
Step-by-Step Breakdown
A) Solving linear equations (one variable)
- Simplify each side: distribute, combine like terms.
- Get variables on one side (add/subtract terms).
- Isolate the variable (multiply/divide).
- Check for special cases:
- No solution: variable cancels and you get false, e.g. 0 = 5.
- Infinitely many solutions: you get true, e.g. 0 = 0.
Mini-example: Solve 3(x-2)=2x+5
- Distribute: 3x-6=2x+5
- Subtract 2x: x-6=5
- Add 6: x=11
B) Solving linear inequalities
- Solve like an equation.
- Flip the inequality only if you multiply/divide by a negative.
- Write in interval form if asked.
Mini-example: Solve -2x+1 \le 7
- Subtract 1: -2x \le 6
- Divide by -2 (flip!): x \ge -3
SAT trap: Forgetting to flip the sign when dividing by a negative.
C) Solving systems of linear equations
Use substitution or elimination.
Elimination steps:
- Align equations as ax+by=c format.
- Multiply one/both equations so a variable’s coefficients are opposites.
- Add/subtract to eliminate.
- Solve for remaining variable.
- Back-substitute.
Mini-example:
Solve
x+y=9
x-y=1
Add: 2x=10 \Rightarrow x=5, then y=4.
D) Solving quadratics
Quadratics appear as ax^2+bx+c=0.
Method choice (fast decision):
- If it factors nicely, factor.
- If it’s already a(x-h)^2+k, use vertex form.
- If not factorable, use the quadratic formula.
Factoring steps:
- Move everything to one side: =0.
- Factor.
- Set each factor to zero.
Mini-example: x^2-5x+6=0
Factor: (x-2)(x-3)=0 so x=2 or x=3.
E) Rational equations (variables in denominators)
- State restrictions: denominator \ne 0.
- Multiply both sides by the LCD (least common denominator).
- Solve the resulting equation.
- Check for extraneous solutions (plug back).
Mini-example: Solve \frac{2}{x-1}=3
- Restriction: x \ne 1
- Multiply: 2=3(x-1)=3x-3
- 3x=5 \Rightarrow x=\frac{5}{3} (valid)
F) Absolute value equations/inequalities
Equation rule:
|A|=b (with b\ge 0) becomes A=b **or** A=-b.
Inequality rules:
- |A|
- |A|>b becomes A>b **or** A
Mini-example: |2x-1|=5
- 2x-1=5 \Rightarrow x=3
- 2x-1=-5 \Rightarrow x=-2
G) Exponential equations (SAT-level)
Often solvable by rewriting to a common base.
Mini-example: Solve 2^{x+1}=8
Rewrite 8=2^3, so x+1=3 \Rightarrow x=2.
Key Formulas, Rules & Facts
Algebra + equation-solving essentials
| Formula / Rule | When to use | Notes / pitfalls |
|---|---|---|
| a(b+c)=ab+ac | Distribute | Don’t forget to distribute to every term |
| ab+ac=a(b+c) | Factor (GCF) | Always check for a GCF first |
| (x+m)(x+n)=x^2+(m+n)x+mn | Multiply binomials | Useful to reverse when factoring |
| x^2-y^2=(x-y)(x+y) | Difference of squares | Common fast factor on SAT |
| (x\pm y)^2=x^2\pm 2xy+y^2 | Perfect square trinomials | Spot patterns for fast factoring |
| If ab=0, then a=0 or b=0 | Zero product property | Only works when product equals 0 |
| \frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc | Proportions | Ensure b\ne 0 and d\ne 0 |
Exponents & radicals
| Rule | When to use | Notes / pitfalls |
|---|---|---|
| a^m\cdot a^n=a^{m+n} | Multiply same base | Bases must match |
| \frac{a^m}{a^n}=a^{m-n} | Divide same base | If m |
| (a^m)^n=a^{mn} | Power of a power | Common place to slip |
| (ab)^n=a^n b^n | Distribute exponent | Works for multiplication |
| \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} | Power of a fraction | Need b\ne 0 |
| a^{-n}=\frac{1}{a^n} | Negative exponents | Means “reciprocal” |
| \sqrt{a}\sqrt{b}=\sqrt{ab} (for a,b\ge 0) | Multiply radicals | Don’t combine across plus: \sqrt{a}+\sqrt{b} stays |
| \sqrt{a^2}=|a| | Simplifying radicals | SAT may test absolute value nuance |
Linear functions & lines
| Formula | When to use | Notes |
|---|---|---|
| m=\frac{y_2-y_1}{x_2-x_1} | Slope between two points | Vertical line: x_2=x_1 (undefined slope) |
| y=mx+b | Line form | b is y-intercept |
| y-y_1=m(x-x_1) | Point-slope form | Great when you have slope + a point |
| Standard form Ax+By=C | Systems / intercepts | Slope is -\frac{A}{B} (if B\ne 0) |
| Parallel lines: m_1=m_2 | Identify parallel | Same slope |
| Perpendicular: m_1m_2=-1 | Identify perpendicular | Negative reciprocal slopes |
Coordinate geometry
| Formula | When to use | Notes |
|---|---|---|
| Distance: d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} | Segment length | Comes from Pythagorean theorem |
| Midpoint: \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) | Midpoint of segment | Average the coordinates |
| Circle: (x-h)^2+(y-k)^2=r^2 | Circle graph/equation | Center (h,k), radius r |
Quadratics (must-know relationships)
| Formula / fact | When to use | Notes |
|---|---|---|
| Quadratic formula: x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} | Solve any quadratic | Watch sign of b carefully |
| Discriminant: \Delta=b^2-4ac | Number of real solutions | \Delta>0 two, \Delta=0 one, \Delta |
| Vertex: for ax^2+bx+c, x=\frac{-b}{2a} | Find axis of symmetry | Then plug in for y |
| Vertex form: a(x-h)^2+k | Graph transformations | Vertex at (h,k) |
Percent, rate, and growth equations
| Equation | When to use | Notes |
|---|---|---|
| Percent change: \frac{\text{new}-\text{old}}{\text{old}}\times 100\% | Increase/decrease | “Of” usually means multiply |
| Simple interest: I=Prt | Interest problems | r as decimal, t in years |
| Exponential growth/decay: A=A_0(1\pm r)^t | Repeated percent change | Use -$ for decay, +$ for growth |
| Average speed: v=\frac{d}{t} | Motion problems | Total avg speed is \frac{\text{total }d}{\text{total }t} |
Examples & Applications
1) Rearranging a formula (literal equations)
Problem style: Solve for a variable in a given formula.
Given V=\frac{1}{3}\pi r^2 h, solve for h.
- Multiply both sides by 3: 3V=\pi r^2 h
- Divide by \pi r^2: h=\frac{3V}{\pi r^2}
Key insight: Treat it like isolating x—just keep operations balanced.
2) System from a word problem
A theater sold 100 tickets. Adult tickets cost \$12 and student tickets cost \$8. Total revenue was \$1040. How many student tickets?
Let a = adult, s = student.
- Count: a+s=100
- Revenue: 12a+8s=1040
Eliminate: multiply first equation by 8:
8a+8s=800
Subtract from revenue equation:
(12a+8s)-(8a+8s)=1040-800 \Rightarrow 4a=240 \Rightarrow a=60
So s=40.
Key insight: “Total number” + “total value” is almost always a 2-equation system.
3) Quadratic from geometry (area)
A rectangle has area 48 and length x+2 and width x-2. Find x.
Set up: (x+2)(x-2)=48
Use difference of squares: x^2-4=48
So x^2=52 \Rightarrow x=\pm\sqrt{52}=\pm 2\sqrt{13}.
SAT reality check: If x is a dimension parameter, you may need **positive only** (and also ensure x-2>0). So x=2\sqrt{13} works.
4) Inequality + interval solution
Solve and graph: 2(1-x)>x+4
- Distribute: 2-2x>x+4
- Subtract 2: -2x>x+2
- Subtract x: -3x>2
- Divide by -3 (flip): x< -\frac{2}{3}
Key insight: The only “special move” is flipping the sign when dividing by a negative.
Common Mistakes & Traps
Forgetting to distribute a negative
- Wrong: -(x-3)=-x-3
- Right: -(x-3)=-x+3
- Fix: Treat -1 as the multiplier and distribute carefully.
Combining unlike terms
- Wrong: 2x+3=5x
- Why wrong: 3 isn’t an x-term.
- Fix: Only combine terms with the exact same variable part.
Illegal canceling across addition
- Wrong: \frac{x+2}{x}=\frac{2}{1}
- Why wrong: You can only cancel factors, not terms in a sum.
- Fix: Factor first if possible.
Not flipping the inequality when dividing by a negative
- Wrong: -2x
Dropping solutions when solving quadratics
- Wrong: Taking only the + in \pm or missing the second factor.
- Fix: For x^2=9, write x=\pm 3 unless context restricts.
Extraneous solutions in rational/absolute value equations
- What happens: Multiplying by a variable expression can introduce invalid answers.
- Fix: State restrictions (like x\ne 1) and plug solutions back.
Misreading function notation
- Mistake: Thinking f(x+2) equals f(x)+2.
- Fix: Replace the entire input: if f(x)=x^2, then f(x+2)=(x+2)^2.
Sign errors in the quadratic formula
- Common slip: Using -b incorrectly when b is negative.
- Fix: If b=-5, then -b=5. Put parentheses: -(-5)=5.
Memory Aids & Quick Tricks
| Trick / mnemonic | What it helps you remember | When to use |
|---|---|---|
| “Same base, add/subtract exponents” | a^m a^n=a^{m+n} and a^m/a^n=a^{m-n} | Simplifying exponent expressions |
| “SOH-CAH-TOA” | \sin=\frac{\text{opp}}{\text{hyp}} etc. | Only if trig appears (rare), but can help with right-triangle ratios |
| “FOIL” | Multiply (a+b)(c+d) | Expanding binomials (though distributing is safer) |
| “Flip when negative” | Inequality sign flips when dividing/multiplying by negative | Inequalities |
| “Circle form = Center/Radius” | (x-h)^2+(y-k)^2=r^2 gives center (h,k) | Circle equation questions |
| “Axis is -b/2a” | Quick vertex x-coordinate | Quadratic graphs/vertex/maximum-minimum |
Quick Review Checklist
- You can distribute, combine like terms, and factor (GCF, difference of squares, perfect square patterns).
- You know exponent rules, including a^{-n}=\frac{1}{a^n} and \sqrt{a^2}=|a|.
- When solving equations, you isolate the variable and watch for no solution vs infinite solutions.
- For inequalities, you flip the sign when multiplying/dividing by a negative.
- For systems, you can do elimination quickly and interpret solutions.
- For quadratics, you can factor or use x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} and use \Delta=b^2-4ac to predict roots.
- For rational equations, you state restrictions and check for extraneous answers.
- You can use line and coordinate formulas: m=\frac{y_2-y_1}{x_2-x_1}, d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}, midpoint, and (x-h)^2+(y-k)^2=r^2.
You don’t need new tricks—just clean algebra and careful sign handling.