PSAT 10 Math Formula & Cheat Sheet

What You Need to Know

You’re aiming for two things on PSAT 10 Math: (1) recognize the type of problem fast, and (2) deploy the right formula/identity/setup with minimal algebra errors. The test leans heavily on algebra (linear, systems, quadratics), functions, ratios/percent, data & probability, and core geometry.

Big idea: most questions are not about memorizing obscure formulas. They’re about using a small set of formulas + clean rearranging + reading graphs/tables carefully.

Critical reminder: The PSAT provides some geometry formulas (areas/volumes, right-triangle facts). You still need to know the rest (slope, midpoint, exponent rules, quadratic facts, etc.) without relying on a provided sheet.

Step-by-Step Breakdown

1) Linear equation / inequality (one variable)

Distribute and combine like terms.
Move variable terms to one side, constants to the other.
Divide to isolate the variable.
For inequalities: if you multiply/divide by a negative, flip the inequality sign.

Mini example:
Solve $3(2x-5)\le 2x+7$

Distribute: $6x-15\le 2x+7$
Subtract $2x$ : $4x-15\le 7$
Add $15$ : $4x\le 22$
Divide by $4$ : $x\le \frac{11}{2}$

2) Linear system (two equations)

Pick a method:

Elimination if coefficients line up (or can be made to line up fast).
Substitution if one equation is already solved for a variable.

Elimination steps:

Multiply one/both equations to create opposite coefficients.
Add/subtract equations to eliminate a variable.
Solve for the remaining variable.
Back-substitute.
(If asked) interpret: intersection point, no solution (parallel), infinitely many (same line).

3) Quadratics (most common pathways)

Put in standard form $ax^2+bx+c=0$ .
Try factoring first (fastest).
If not factorable quickly, use quadratic formula.
Use discriminant $b^2-4ac$ to predict number of real solutions.

Mini example (factoring):
Solve $x^2-5x+6=0$
Factor: $(x-2)(x-3)=0$ so $x=2$ or $x=3$ .

4) Word problems (rates, ratios, percent)

Define variables with units.
Write an equation using a known structure:
- Percent: $\text{part}=\text{percent}\times\text{whole}$ (percent as a decimal)
- Rate: $\text{distance}=\text{rate}\times\text{time}$
- Work: $\text{work rate} = \frac{1}{\text{time}}$ (combine rates by adding)
Solve algebraically, then check if the answer makes sense (sign, size, units).

5) Geometry “target quantity” approach

Sketch and label everything (especially radii, diameters, heights).
Identify the exact target (area, perimeter, volume, angle, side length).
Choose the formula and solve for missing pieces (often via similar triangles or Pythagorean theorem).
Confirm you used consistent units.

Key Formulas, Rules & Facts

A) Algebra essentials (high-frequency)

Formula / Rule	When to use	Notes
$ax+b=c \Rightarrow x=\frac{c-b}{a}$	Solve linear equations	Keep signs straight.
$a(b+c)=ab+ac$	Distribute	Common source of mistakes.
$\|x-a\|=b$	Absolute value equations	Requires $b\ge 0$ ; solutions $x=a\pm b$ .
$\|x-a\|<b$	Absolute value inequalities	Turns into $a-b<x<a+b$ .
$m=\frac{y_2-y_1}{x_2-x_1}$	Slope	Vertical line has undefined slope.
$y-y_1=m(x-x_1)$	Point-slope form	Convert to slope-intercept if needed.
$y=mx+b$	Slope-intercept form	$b$ is y-intercept.
$x=h$ , $y=k$	Vertical / horizontal lines	Vertical lines: same $x$ . Horizontal: same $y$ .
$\text{average rate of change}=\frac{f(b)-f(a)}{b-a}$	Function graphs/tables	Same as slope between two points.

B) Exponents, radicals, and expressions

Rule	Use	Notes
$a^m\cdot a^n=a^{m+n}$	Multiply same base	Add exponents.
$\frac{a^m}{a^n}=a^{m-n}$	Divide same base	Subtract exponents.
$(a^m)^n=a^{mn}$	Power of a power	Multiply exponents.
$(ab)^n=a^n b^n$	Power of product	Also works for quotients.
$a^0=1$	Simplify	For $a\ne 0$ .
$a^{-n}=\frac{1}{a^n}$	Negative exponents	Rewrite to avoid negatives.
$\sqrt{ab}=\sqrt{a}\sqrt{b}$	Simplify radicals	For $a\ge 0$ and $b\ge 0$ (real numbers).

C) Quadratics and polynomials

Formula / Fact	When to use	Notes
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$	Solve any quadratic	Works always (real/complex).
$\Delta=b^2-4ac$	# of real solutions	$\Delta>0$ two; $\Delta=0$ one; $\Delta<0$ none (real).
If $r$ is a root of $ax^2+bx+c$	Link roots to factors	Then $a(x-r)$ is a factor (often used with given root).
Vertex of $y=ax^2+bx+c$ has $x=-\frac{b}{2a}$	Find max/min	Plug back in for $y$ .

D) Functions (what PSAT loves testing)

Concept	What it means	Quick notes
$f(x)$	Output when input is $x$	Read as “f of x.”
Domain	Allowed inputs	Watch denominators and square roots.
Range	Possible outputs	Often from graph or constraints.
$f(a)$	Substitute $a$ for $x$	Most common function mistake is plugging in wrong.
Linear vs exponential	Constant difference vs constant ratio	Linear: add same amount; exponential: multiply by same factor.

E) Ratios, proportions, and percent

Formula / Rule	When to use	Notes
$\frac{a}{b}=\frac{c}{d} \Rightarrow ad=bc$	Proportions	Cross-multiply carefully.
$\text{part}=\text{percent}\times\text{whole}$	Percent problems	Convert percent to decimal.
$\text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}$	Growth/decline	Multiply by $100\%$ only if asked for percent.
Weighted average	Mix problems	$\frac{w_1x_1+w_2x_2}{w_1+w_2}$ with weights.

F) Geometry (must-know)

1) Triangles and right triangles

Formula / Fact	When to use	Notes
$A=\frac{1}{2}bh$	Triangle area	Height is perpendicular to base.
$a^2+b^2=c^2$	Right triangles	$c$ is hypotenuse (longest side).
$45\text{-}45\text{-}90$ ratios	Special right triangle	Legs $x,x$ ; hypotenuse $x\sqrt{2}$ .
$30\text{-}60\text{-}90$ ratios	Special right triangle	Short $x$ ; long $x\sqrt{3}$ ; hypotenuse $2x$ .
Similar triangles	Scale factors	Corresponding sides proportional.

2) Circles

Formula	When to use	Notes
$C=2\pi r$	Circumference	Also $C=\pi d$ with $d=2r$ .
$A=\pi r^2$	Area	Keep exact $\pi$ unless asked for decimal.
Arc length / sector area	If given fraction of circle	Use fraction of $2\pi r$ or $\pi r^2$ .

3) Common area and volume

Shape	Formula	Notes
Rectangle	$A=lw$	Perimeter $P=2l+2w$ .
Parallelogram	$A=bh$	Height is perpendicular.
Trapezoid	$A=\frac{1}{2}(b_1+b_2)h$	Bases are parallel sides.
Rectangular prism	$V=lwh$	Surface area may appear; count faces.
Cylinder	$V=\pi r^2 h$	“Base area times height.”
Cone	$V=\frac{1}{3}\pi r^2 h$	One-third of cylinder.
Sphere	$V=\frac{4}{3}\pi r^3$	Radius cubed.

4) Coordinate geometry

Formula	When to use	Notes
$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$	Distance	Also derived from Pythagorean theorem.
$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$	Midpoint	Often used in geometry-in-the-plane.

G) Data, stats, and probability

Concept / Formula	When to use	Notes
Mean	$\bar{x}=\frac{\text{sum}}{n}$	If one value changes, mean changes predictably.
Median	Middle value	Sort first. If even $n$ , average middle two.
Range	$\text{max}-\text{min}$	Quick spread measure.
Probability	$P(A)=\frac{\text{favorable}}{\text{total}}$	Must be equally likely outcomes.
Complement	$P(A^c)=1-P(A)$	Great when “at least one” is hard directly.
Independent events	$P(A\cap B)=P(A)P(B)$	“And” with independence.
Conditional probability	$P(A\|B)=\frac{P(A\cap B)}{P(B)}$	“Given” reduces sample space.

Examples & Applications

Example 1: Function interpretation (table/graph style)

If $f(x)=2x^2-3x+1$ , find $f(-2)$ .

Substitute:
$f(-2)=2(-2)^2-3(-2)+1$
$f(-2)=2\cdot 4+6+1=15$

Exam variation: They may hide this in context (“profit at time $t$ ”), but it’s the same substitution.

Example 2: System as intersection

Solve:
$y=2x+1$
$y=-x+7$

Set equal (substitution):
$2x+1=-x+7$
$3x=6$ so $x=2$ .
Then $y=2(2)+1=5$ .

Solution is $\left(2,5\right)$ .

Exam variation: They might ask “How many solutions?” (one, none, infinite) by comparing slopes/intercepts.

Example 3: Percent change

A price increases from $80$ to $92$ . Find the percent increase.

$\text{percent change}=\frac{92-80}{80}=\frac{12}{80}=0.15$
So the increase is $15\%$ .

Exam variation: Reverse percent: “after a $15\%$ increase the price is $92$ , what was original?” Use $92=1.15\times \text{original}$ .

Example 4: Geometry with special right triangle

A right triangle has a $30^\circ$ angle and hypotenuse $10$ . Find the shorter leg.

In a $30\text{-}60\text{-}90$ triangle, hypotenuse $=2x$ where $x$ is the short leg.
So $2x=10 \Rightarrow x=5$ .

Exam variation: They may embed this in an equilateral triangle split in half, or in a complex diagram.

Common Mistakes & Traps

Sign errors when distributing or moving terms
- What goes wrong: you forget to distribute a negative, e.g., $-(x-3)=-x-3$ (not $-x+3$ ).
- Fix: rewrite subtraction as adding a negative: $-(x-3)=(-1)(x-3)$ .
Forgetting to flip an inequality
- What goes wrong: you divide both sides by a negative and keep the inequality direction.
- Fix: highlight the step where you multiply/divide by a negative; flip the symbol immediately.
Mixing up slope with intercepts
- What goes wrong: you treat $b$ (y-intercept) like slope in $y=mx+b$ .
- Fix: remember slope is the “per 1” change: $m=\frac{\Delta y}{\Delta x}$ .
Using Pythagorean theorem with the wrong side as $c$
- What goes wrong: you plug the hypotenuse into $a$ or $b$ .
- Fix: hypotenuse is always opposite the right angle and is the longest side; label it before calculating.
Treating percent like a whole number instead of a decimal
- What goes wrong: using $15$ instead of $0.15$ in $\text{part}=\text{percent}\times\text{whole}$ .
- Fix: convert: $15\%=0.15$ , $7\%=0.07$ , $120\%=1.2$ .
Misreading “of” vs “off” and “more than” vs “times as much”
- What goes wrong: “ $30\%$ off” means multiply by $0.70$ , not $1.30$ .
- Fix: discount: $\text{new}=(1-\text{rate})\times \text{old}$ ; markup: $\text{new}=(1+\text{rate})\times \text{old}$ .
Quadratic factoring slips
- What goes wrong: you find numbers that multiply to $ac$ but don’t add to $b$ .
- Fix: check by expanding your factors quickly before committing.
Probability sample space mismatch
- What goes wrong: you count favorable outcomes from one situation but divide by a different total.
- Fix: define the sample space in words first (especially for “given” problems).

Memory Aids & Quick Tricks

Trick / Mnemonic	Helps you remember	When to use
“SOH-CAH-TOA”	Trig ratios $\sin$ , $\cos$ , $\tan$	If basic trig appears (some PSAT problems do).
“Rise over run”	$m=\frac{\Delta y}{\Delta x}$	Slope from two points or graphs.
“Keep-Change-Flip”	Dividing fractions: $\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}$	Rational expression simplification.
“30-60-90: 1, \sqrt{3}, 2”	Side ratios	Fast geometry side finding.
“45-45-90: 1, 1, \sqrt{2}”	Side ratios	Squares/diagonals show up a lot.
“New = Old(1 \pm r)”	Percent increase/decrease	Any percent change setup.
“Vertex x is -b over 2a”	$x=-\frac{b}{2a}$	Max/min and graph features of quadratics.
“Complement: 1 minus”	$P(A^c)=1-P(A)$	“At least one” probability.

Quick Review Checklist

You can solve linear equations/inequalities cleanly and flip inequalities when needed.
You can find slope using $\frac{y_2-y_1}{x_2-x_1}$ and write equations of lines.
You can evaluate functions $f(a)$ and interpret average rate of change.
You can solve systems by substitution/elimination and recognize 1 vs 0 vs infinite solutions.
You can factor simple quadratics and use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ when factoring fails.
You can use percent/ratio setups correctly (convert percents to decimals).
You know core geometry formulas (areas/volumes) and right-triangle facts, including special triangles.
You can compute mean/median/range and basic probabilities, including complements.

You’ve got the tools; now it’s just clean setup and careful arithmetic under time pressure.