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)

  1. Simplify each side (distribute, combine like terms).

  2. Move variable terms to one side, constants to the other.

  3. Isolate the variable (divide by coefficient).

  4. 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

  1. Solve like an equation.

  2. If you multiply/divide by a negative, flip the inequality sign.

  3. 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:

  1. Substitution if a variable is already isolated or easy to isolate.

  2. Elimination if coefficients line up (or can be made to).

  3. 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:

  1. Factor (fastest if it factors nicely).

  2. Quadratic formula if factoring is messy.

  3. 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 $|A|$-b.

  • If $|A|>b$, then $A>b$ or $A<-b$.

Mini-example: $|2x-3|=7$

  • $2x-3=7\Rightarrow x=5$

  • $2x-3=-7\Rightarrow x=-2$

F) Rational equations (variables in denominators)

  1. State restrictions: denominators $\ne 0$.

  2. Multiply both sides by the LCD (least common denominator).

  3. Solve the resulting equation.

  4. Check against restrictions.

G) Radical equations

  1. Isolate the radical.

  2. Square both sides.

  3. Solve.

  4. Check in the original (squaring often introduces extraneous roots).

H) Rearranging formulas (solve for a variable)

  1. Treat it like an equation-solving problem.

  2. Undo operations in reverse order.

  3. 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

  1. 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 ($<$ becomes $>$, etc.).

  2. 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.

  3. Dropping parentheses with negatives

    • Wrong: $-(x-3)=-x-3$ (should be $-x+3$).

    • Fix: distribute the negative as multiplying by $-1$.

  4. 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).

  5. 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.

  6. Mixing up line formulas (slope vs intercept)

    • Wrong: thinking $b$ in $y=mx+b$ is slope.

    • Fix: $m$ is slope, $b$ is y-intercept.

  7. Sign errors in the quadratic formula

    • Wrong: using $\frac{b\pm\sqrt{b^2-4ac}}{2a}$.

    • Fix: it’s $-b$ on top: $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

  8. 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: $\Delta>0,=0,<0$

“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.