Math Formulas/Equations
What You Need to Know
On SAT Math, a huge chunk of points comes from recognizing the right equation/formula quickly and then manipulating it cleanly (solve, substitute, rearrange, compare forms). You’re rarely doing “hard math”; you’re doing accurate algebra under time pressure.
The core skill
You must be able to:
- Translate words to equations (e.g., “is”, “of”, “more than”, “per”, “at least”).
- Solve equations/inequalities (linear, quadratic, absolute value, rational, radical, exponential basics).
- Rearrange formulas (solve for a variable).
- Recognize equivalent forms (especially for quadratics, lines, exponent rules).
Critical reminder: If you square both sides, multiply by a variable expression, or clear denominators, you can create extraneous solutions. Always check in the original equation.
Step-by-Step Breakdown
A) Solving linear equations (1 variable)
- Simplify each side (distribute, combine like terms).
- Move variable terms to one side, constants to the other.
- Isolate the variable (divide by coefficient).
- Check if the problem came from fractions/absolute values (quick plug-in check).
Mini-example: Solve $3(2x-1)=5x+7$
- Distribute: $6x-3=5x+7$
- Subtract $5x$: $x-3=7$
- Add 3: $x=10$
B) Solving linear inequalities
- Solve like an equation.
- If you multiply/divide by a negative, flip the inequality sign.
- If asked for solutions on a number line, use interval notation or inequality form.
Mini-example: $-2x+5\ge 9$
- $-2x\ge 4$
- Divide by $-2$ (flip): $x\le -2$
C) Systems of equations (2 variables)
Use the method that looks fastest:
- Substitution if a variable is already isolated or easy to isolate.
- Elimination if coefficients line up (or can be made to).
- Graphing logic if the question is about number of solutions or intersection behavior.
Mini-example (elimination):
\begin{align}
2x+y&=11\
2x-y&=5
\end{align}
Add: $4x=16\Rightarrow x=4$ then $2(4)+y=11\Rightarrow y=3$.
D) Quadratic equations
Common solving tools:
- Factor (fastest if it factors nicely).
- Quadratic formula if factoring is messy.
- Complete the square (also helps convert to vertex form).
Mini-example (factoring): $x^2-5x+6=0\Rightarrow (x-2)(x-3)=0\Rightarrow x=2,3$.
E) Absolute value equations/inequalities
Key identity:
- If $|A|=b$ with $b\ge 0$, then $A=b$ or $A=-b$.
- If $|A|
- If $|A|>b$, then $A>b$ or $A
Mini-example: $|2x-3|=7$
- $2x-3=7\Rightarrow x=5$
- $2x-3=-7\Rightarrow x=-2$
F) Rational equations (variables in denominators)
- State restrictions: denominators $\ne 0$.
- Multiply both sides by the LCD (least common denominator).
- Solve the resulting equation.
- Check against restrictions.
G) Radical equations
- Isolate the radical.
- Square both sides.
- Solve.
- Check in the original (squaring often introduces extraneous roots).
H) Rearranging formulas (solve for a variable)
- Treat it like an equation-solving problem.
- Undo operations in reverse order.
- If the variable appears in multiple terms, factor it out.
Mini-example: Solve $A=\frac{1}{2}bh$ for $h$.
- Multiply by 2: $2A=bh$
- Divide by $b$: $h=\frac{2A}{b}$
Key Formulas, Rules & Facts
Algebra & equation forms
| Formula/Rule | When to use | Notes |
|---|---|---|
| Distributive: $a(b+c)=ab+ac$ | Expand/simplify | Common error: forgetting to distribute negatives |
| Factoring GCF: $ax+ay=a(x+y)$ | Pull out common factor | Helps solve and simplify |
| Difference of squares: $a^2-b^2=(a-b)(a+b)$ | Recognize patterns | Shows up in simplifying/rationalizing |
| Perfect squares: $(a\pm b)^2=a^2\pm 2ab+b^2$ | Expanding/factoring | Middle term sign matches $\pm$ |
| Quadratic standard form: $ax^2+bx+c=0$ | General quadratic | $a\ne 0$ |
| Quadratic formula: x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} | Non-factorable quadratics | Discriminant $\Delta=b^2-4ac$ |
| Vertex form: $y=a(x-h)^2+k$ | Vertex/transformations | Vertex $(h,k)$ |
| Axis of symmetry: $x=\frac{-b}{2a}$ | Quadratic graph features | From $ax^2+bx+c$ |
| Exponent rules: $a^m a^n=a^{m+n}$, $\frac{a^m}{a^n}=a^{m-n}$ | Simplify exponents | $a\ne 0$ for division |
| Power rules: $(a^m)^n=a^{mn}$, $(ab)^n=a^n b^n$ | Simplify | Watch parentheses |
| Negative exponent: $a^{-n}=\frac{1}{a^n}$ | Rewrite | $a\ne 0$ |
| Fractional exponent: $a^{1/n}=\sqrt[n]{a}$ | Convert radicals/exponents | Even roots require $a\ge 0$ in reals |
Linear equations, lines, and coordinate geometry
| Formula/Rule | When to use | Notes |
|---|---|---|
| Slope: m=\frac{y_2-y_1}{x_2-x_1} | Rate of change | Vertical line: undefined slope |
| Slope-intercept: $y=mx+b$ | Graphing/reading line | $b$ is y-intercept |
| Point-slope: $y-y_1=m(x-x_1)$ | Line through point with slope | Great for quick writing |
| Standard form: $Ax+By=C$ | Systems/elimination | Many equivalent forms |
| Parallel lines: $m_1=m_2$ | Relationship questions | Same slope |
| Perpendicular: $m_1m_2=-1$ | Right angles | Negative reciprocal |
| Distance: d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} | Length between points | Pythagorean theorem in plane |
| Midpoint: \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) | Segment midpoint | Common in coordinate geometry |
| Circle: (x-h)^2+(y-k)^2=r^2 | Circle graphs | Center $(h,k)$ radius $r$ |
Geometry essentials (commonly used equations)
| Formula/Rule | When to use | Notes |
|---|---|---|
| Triangle area: A=\frac{1}{2}bh | Any triangle | Height is perpendicular to base |
| Rectangle area: $A=lw$ | Rectangles | |
| Parallelogram area: $A=bh$ | Parallelograms | |
| Trapezoid area: A=\frac{1}{2}(b_1+b_2)h | Trapezoids | Bases are parallel sides |
| Circle circumference: $C=2\pi r$ | Circle perimeter | |
| Circle area: $A=\pi r^2$ | Circles | |
| Arc length: s=\frac{\theta}{360^\circ}(2\pi r) | Degrees given | If radians: $s=r\theta$ |
| Sector area: A=\frac{\theta}{360^\circ}(\pi r^2) | Portion of circle | Degrees version |
| Pythagorean theorem: a^2+b^2=c^2 | Right triangles | $c$ is hypotenuse |
| 45-45-90 triangle | Fast side ratios | $x, x, x\sqrt{2}$ |
| 30-60-90 triangle | Fast side ratios | $x, x\sqrt{3}, 2x$ (short, long, hyp.) |
| Volume prism/cyl: $V=Bh$ | 3D solids | $B$ is base area |
| Cylinder volume: $V=\pi r^2 h$ | Cylinders | Same as $Bh$ |
| Sphere volume: V=\frac{4}{3}\pi r^3 | Spheres | Often tested |
| Cone volume: V=\frac{1}{3}\pi r^2 h | Cones | “One-third of cylinder” |
Data & probability equations
| Formula/Rule | When to use | Notes |
|---|---|---|
| Mean: \bar{x}=\frac{\text{sum}}{n} | Average | Total = mean $\times n$ |
| Median | Middle value | Sort first |
| Percent change: \%\text{change}=\frac{\text{new-old}}{\text{old}}\times100\% | Increase/decrease | Watch sign |
| Probability: P(E)=\frac{\text{favorable}}{\text{total}} | Simple probability | Assume equally likely outcomes |
| Independent events: $P(A\cap B)=P(A)P(B)$ | “and” with independence | Often from replacement |
| Exclusive events: $P(A\cup B)=P(A)+P(B)$ | “or” with no overlap | If overlap exists subtract it |
Examples & Applications
1) Rearranging a formula (classic SAT)
Problem: If $P=2L+2W$, solve for $W$.
- Subtract $2L$: $P-2L=2W$
- Divide by 2: W=\frac{P-2L}{2}
Insight: Keep expressions grouped; don’t distribute unless it helps.
2) Rational equation with restriction
Problem: Solve $\frac{3}{x-1}=2$.
- Restriction: $x\ne 1$
- Multiply: $3=2(x-1)=2x-2$
- $2x=5\Rightarrow x=\frac{5}{2}$ (valid)
Insight: The restriction step prevents illegal answers.
3) Quadratic in disguise (substitution)
Problem: Solve $x^4-5x^2+4=0$.
Let $u=x^2$:
- $u^2-5u+4=0\Rightarrow (u-1)(u-4)=0$
- $u=1$ or $u=4$
- So $x^2=1\Rightarrow x=\pm1$; $x^2=4\Rightarrow x=\pm2$
Insight: Look for “quadratic pattern” in $x^2$, $x^3$, etc.
4) System word problem (set up equations)
Problem: Adult tickets cost $\$12$, student tickets cost $\$8$. Total tickets: 25. Total revenue: $\$244$. How many adult tickets?
Let $a$ = adult, $s$ = student.
- Count: $a+s=25$
- Money: $12a+8s=244$
Substitute $s=25-a$: - $12a+8(25-a)=244$
- $12a+200-8a=244\Rightarrow4a=44\Rightarrow a=11$
Insight: One equation is “how many”; the other is “how much.”
Common Mistakes & Traps
Forgetting to flip an inequality
- Wrong: dividing by a negative and keeping the same sign.
- Fix: any time you multiply/divide by a negative, flip ($
Not checking extraneous solutions
- Happens after squaring both sides or clearing denominators.
- Fix: plug final answers back into the original equation, especially for radicals/rationals.
Dropping parentheses with negatives
- Wrong: $-(x-3)=-x-3$ (should be $-x+3$).
- Fix: distribute the negative as multiplying by $-1$.
Misusing absolute value rules
- Wrong: $|x|=5\Rightarrow x=5$ only.
- Fix: write two cases: $x=5$ or $x=-5$ (when the RHS is positive).
Cancelling terms incorrectly in rational expressions
- Wrong: $\frac{x+2}{x}=\frac{2}{1}$ by “cancelling $x$.”
- Fix: you can cancel factors, not terms. Only cancel when something is multiplied.
Mixing up line formulas (slope vs intercept)
- Wrong: thinking $b$ in $y=mx+b$ is slope.
- Fix: $m$ is slope, $b$ is y-intercept.
Sign errors in the quadratic formula
- Wrong: using $\frac{b\pm\sqrt{b^2-4ac}}{2a}$.
- Fix: it’s $-b$ on top: $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Assuming “or” means add probabilities automatically
- Wrong: adding $P(A)+P(B)$ when events overlap.
- Fix: if overlap is possible: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
Memory Aids & Quick Tricks
| Trick/Mnemonic | Helps you remember | When to use |
|---|---|---|
| FOIL (First, Outer, Inner, Last) | Multiply two binomials | Expanding $(a+b)(c+d)$ |
| “Flip when negative” | Inequality sign flips | Dividing/multiplying inequalities by negatives |
| 30-60-90: $x, x\sqrt{3}, 2x$ | Special right triangle ratios | Geometry with 30°/60°/90° |
| 45-45-90: $x, x, x\sqrt{2}$ | Special right triangle ratios | Squares/diagonals/isosceles right triangles |
| Circle: (x-h)^2+(y-k)^2=r^2 | Center-radius form | Any circle equation question |
| Mean = total ÷ number (so total = mean×n) | Back-solve quickly | “After adding/removing a value, what’s new mean?” |
| Discriminant $\Delta=b^2-4ac$ | # of real roots | Quadratics without fully solving: $\Delta>0,=0, |
| “Clear denominators with LCD” | Rational equations | Fractions in equations |
Quick Review Checklist
- You can rearrange formulas by undoing operations and factoring the variable out.
- You solve inequalities like equations, but flip the sign when dividing/multiplying by a negative.
- For systems, pick the fastest method (substitution/elimination).
- For quadratics, try factoring first; use the quadratic formula when needed.
- For absolute value, split into two cases (or a compound inequality).
- For rational/radical equations, write restrictions and check for extraneous solutions.
- You know the line tools: slope formula, $y=mx+b$, point-slope, parallel/perpendicular rules.
- You recognize the most-used geometry equations: areas, volumes, special right triangles, circle formulas.
One last push: if you keep your algebra clean and always do a quick “does this answer make sense?” check, you’ll catch most SAT traps.