PSAT 8/9 Math — Foundations, Algebra, Data, and Geometry (Taught from Scratch)

Numbers, Ratios, Percents, and Units

A large part of PSAT 8/9 Math is about being fluent with numbers in real contexts—money, time, distance, measurements, and “how much per” comparisons. These problems often look simple, but they test whether you can reason with quantities, keep units consistent, and choose a sensible strategy.

Place value, factors, multiples, and primes

Place value is the value a digit has because of where it sits in a number. Place value matters when you estimate, compare numbers, and interpret decimals.

A factor of a number is a whole number that divides it with no remainder. A multiple of a number is the result of multiplying it by an integer. These ideas show up when simplifying fractions, working with least common denominators, and understanding divisibility.

A prime number is a whole number greater than $1$ with exactly two positive factors: $1$ and itself. Primes matter because every whole number greater than $1$ can be written as a product of primes (prime factorization). That’s useful for simplifying fractions and for reasoning about common factors.

Example (prime factorization to simplify a fraction)

Simplify $\frac{84}{126}$.

1. Factor each number:

$84 = 2^2 \times 3 \times 7$

$126 = 2 \times 3^2 \times 7$

2. Cancel common factors:

$\frac{84}{126} = \frac{2^2 \times 3 \times 7}{2 \times 3^2 \times 7} = \frac{2}{3}$

A common mistake is canceling terms that are added rather than multiplied. You can only cancel factors, not parts of a sum.

Fractions, decimals, and percents (and why conversions matter)

Fractions, decimals, and percents are three ways to represent the same idea: parts of a whole. The test often checks whether you can switch forms flexibly.

• A percent means “out of $100$.” So $35\%$ means $\frac{35}{100}$.
• To convert percent to decimal, divide by $100$.
• To convert decimal to percent, multiply by $100$.

Example (percent to fraction and decimal)

Convert $18\%$.

$18\% = \frac{18}{100} = \frac{9}{50}$

$18\% = 0.18$

A common error is writing $18\%$ as $18$ (forgetting percent is a scaled value).

Ratios and rates

A ratio compares two quantities by division. For example, a ratio of $3$ to $2$ can be written as:

$3:2$

$\frac{3}{2}$

A rate is a ratio with different units, like $\text{miles per hour}$. A **unit rate** is a rate per $1$ unit, like $\text{dollars per 1 pound}$. Unit rates help you compare options.

Example (unit rate comparison)

Option A: $12$ stickers cost $\$4.80$.

Option B: $20$ stickers cost $\$7.00$.

Compute cost per sticker.

Option A:

$\frac{4.80}{12} = 0.40$

Option B:

$\frac{7.00}{20} = 0.35$

Option B is cheaper per sticker.

A common mistake is comparing totals instead of unit rates when the amounts differ.

Proportions and scaling

A proportion is an equation stating two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

Proportions model scaling, similar figures, recipes, map distances, and speed problems.

A reliable method is cross-multiplication (valid when $b \neq 0$ and $d \neq 0$):

$\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc$

Example (recipe scaling)

A recipe uses $3$ cups of flour for $8$ muffins. How much flour for $20$ muffins?

Set up a proportion with matching units:

$\frac{3}{8} = \frac{x}{20}$

Cross-multiply:

$3 \times 20 = 8x$

$60 = 8x$

$x = 7.5$

So you need $7.5$ cups.

A common mistake is mixing up what corresponds (for example, putting muffins on top in one ratio and on bottom in the other).

Percent change (increase and decrease)

Percent change compares a change to the original amount:

$\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%$

If the result is positive, it’s a percent increase; if negative, a percent decrease.

Another approach many students find easier is “multiply by a percent multiplier”:

• Increase by $p\%$: multiply by $1 + \frac{p}{100}$
• Decrease by $p\%$: multiply by $1 - \frac{p}{100}$

Example (discount)

A jacket costs $\$60$ and is discounted by $25\%$. New price:

$60 \times \left(1 - \frac{25}{100}\right) = 60 \times 0.75 = 45$

The new price is $\$45$.

A frequent mistake is subtracting $25$ instead of $25\%$ of the price.

Units and conversions (dimensional analysis)

PSAT 8/9 problems often hide difficulty in units. The safest method is to multiply by conversion factors that equal $1$.

Example (unit conversion)

Convert $3.5$ hours to minutes.

Use $60$ minutes per hour:

$3.5 \times 60 = 210$

So $3.5$ hours is $210$ minutes.

Common mistakes include converting in the wrong direction (multiplying when you should divide) and mixing units inside a ratio.

Exam Focus

• Typical question patterns:
  • Multi-step word problems involving percent discount/tax/tip, or percent change over time.
  • Unit-rate comparisons (best buy) from tables or short descriptions.
  • Proportions with scaling (recipes, maps, similar “per” relationships).
• Common mistakes:
  • Setting up ratios with mismatched order (swapping what goes on top/bottom).
  • Treating $x\%$ as $x$ instead of $\frac{x}{100}$.
  • Dropping or mixing units, especially in rate problems.

Expressions and the Language of Algebra

Algebra starts with learning to describe patterns and quantities using symbols. An algebraic expression is like a “math phrase”—it has numbers, variables, and operations, but no equals sign. An equation is a statement that two expressions are equal.

Variables and what expressions mean

A variable (like $x$) stands for an unknown number or a quantity that can change. Variables let you write general rules.

For example, “three more than a number” becomes:

$x + 3$

It’s important to translate carefully. Words like “more than” and “less than” can flip order.

• “$5$ less than $x$” is:

$x - 5$

• “$5$ less than twice $x$” is:

$2x - 5$

A common mistake is writing $5 - x$ because you see the word “less.” The phrase “less than” tells you the subtraction order.

Like terms and simplifying

Like terms are terms with the same variable part (same variables to the same powers). You can combine like terms because they represent the same kind of quantity.

For example:

$3x + 5x = 8x$

But:

$3x + 5y$

cannot be combined because $x$ and $y$ are different quantities.

Example (simplify)

Simplify:

$4x + 7 - 2x + 3$

Group like terms:

$(4x - 2x) + (7 + 3) = 2x + 10$

Common mistakes include combining unlike terms (turning $x + x^2$ into $2x^2$, which is not valid).

The distributive property

The distributive property connects multiplication and addition:

$a(b + c) = ab + ac$

It matters because it’s the engine behind expanding expressions, factoring, and solving equations.

Example (expand)

Expand:

$3(2x - 5)$

Multiply $3$ by each term:

$3 \times 2x + 3 \times (-5) = 6x - 15$

A common mistake is distributing to only the first term.

Factoring as “reverse distribution”

Factoring rewrites an expression as a product. This helps when solving equations (especially quadratic ones later) and simplifying algebraic forms.

The most common factoring move on PSAT 8/9 is taking out the greatest common factor (GCF).

Example (GCF factoring)

Factor:

$12x + 18$

The GCF of $12$ and $18$ is $6$:

$12x + 18 = 6(2x + 3)$

A common mistake is factoring out a number that is not actually common to every term.

Exponents (powers) and their rules

An exponent tells you how many times to multiply a base by itself.

$2^4 = 2 \times 2 \times 2 \times 2$

Exponent rules help you simplify expressions efficiently. The key idea is that exponents track repeated multiplication.

• Product of powers (same base):

$a^m \times a^n = a^{m+n}$

• Quotient of powers (same base, $a \neq 0$):

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

• Power of a power:

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

• Zero exponent (for $a \neq 0$):

$a^0 = 1$

• Negative exponent (for $a \neq 0$):

$a^{-n} = \frac{1}{a^n}$

Example (simplify with exponents)

Simplify:

$\frac{2^5 \times 2^3}{2^4}$

Combine numerator powers:

$2^{5+3} = 2^8$

Divide by $2^4$:

$\frac{2^8}{2^4} = 2^{8-4} = 2^4 = 16$

A frequent mistake is adding exponents when bases are different, like treating $2^3 \times 3^3$ as $5^3$ (not true). What is true is:

$2^3 \times 3^3 = (2 \times 3)^3$

Square roots and radicals

A square root is the number that squares to give the original value. For example, because $7^2 = 49$, we have:

$\sqrt{49} = 7$

Many PSAT 8/9 radical questions involve recognizing perfect squares and estimating when a root is not perfect.

Example (simplify a radical)

Simplify:

$\sqrt{72}$

Break $72$ into a perfect square times something else:

$72 = 36 \times 2$

Then:

$\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$

A common mistake is thinking $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$. That is not generally true.

Exam Focus

• Typical question patterns:
  • Simplifying expressions by combining like terms and distributing.
  • Translating word phrases into algebraic expressions.
  • Basic exponent-rule simplification and identifying equivalent expressions.
• Common mistakes:
  • Misreading “less than” and reversing subtraction order.
  • Distributing incorrectly (especially with negative numbers like $-(x-4)$).
  • Using exponent rules when bases don’t match.

Solving Linear Equations and Inequalities

A linear equation is an equation where the variable has exponent $1$ (no squares, roots, or products like $xy$). Linear equations model constant-rate situations: the total changes by the same amount each time.

The core skill is to isolate the variable using inverse operations while keeping the equation balanced.

What it means to “solve” an equation

To solve means to find the value(s) of the variable that make the equation true. You can think of the equals sign as a balance: whatever you do to one side, you must do to the other.

One-step and multi-step equations

When you solve, you usually undo operations in reverse order.

Example (multi-step)

Solve:

$5x - 7 = 18$

1. Add $7$ to both sides:

$5x = 25$

2. Divide both sides by $5$:

$x = 5$

A common mistake is subtracting $7$ instead of adding it.

Equations with variables on both sides

These test whether you can “collect” variable terms on one side.

Example (variables on both sides)

Solve:

$3x + 4 = x + 18$

1. Subtract $x$ from both sides:

$2x + 4 = 18$

2. Subtract $4$:

$2x = 14$

3. Divide by $2$:

$x = 7$

A common mistake is moving terms across the equals sign and changing signs inconsistently. If you prefer, stick to “add/subtract the same amount on both sides” rather than “moving.”

Literal equations (solving for a variable)

Sometimes you solve for a variable in terms of others—common in formulas.

Example (solve for a variable)

Solve for $h$:

$A = bh$

Divide both sides by $b$ (assuming $b \neq 0$):

$h = \frac{A}{b}$

Inequalities and solution sets

An inequality compares expressions using symbols like $<$, $\leq$, $>$, $\geq$. Solving an inequality is similar to solving an equation—except for one special rule.

If you multiply or divide both sides by a negative number, you must flip the inequality sign.

Example (inequality with sign flip)

Solve:

$-2x + 1 > 9$

1. Subtract $1$:

$-2x > 8$

2. Divide by $-2$ and flip the sign:

$x < -4$

Students often forget to flip the sign, which gives the opposite set of solutions.

Interpreting solutions on a number line

Inequalities describe ranges.

• $x < 3$ means all numbers less than $3$ (open circle at $3$).
• $x \leq 3$ includes $3$ (closed circle at $3$).

Modeling with linear equations

Modeling means translating a situation into an equation and interpreting the answer back in context.

Example (ticket cost model)

A movie ticket costs $\$9$ plus a one-time online fee of $\$2$. Write an expression for the total cost for $t$ tickets and find the cost for $t = 4$.

The variable part is $9t$ and the fixed fee is $2$:

$9t + 2$

For $t = 4$:

$9(4) + 2 = 36 + 2 = 38$

So the total is $\$38$.

A common mistake is multiplying the fee by $t$ even though it is one-time.

Exam Focus

• Typical question patterns:
  • Solve multi-step equations, including parentheses and fractions.
  • Solve inequalities and interpret them (often with a number line or word context).
  • Write an equation from a word problem with a fixed amount plus a per-item rate.
• Common mistakes:
  • Distributing incorrectly through parentheses, especially with negatives.
  • Forgetting to flip the inequality when dividing by a negative.
  • Losing track of what the variable represents in a word problem.

Linear Functions, Graphs, and Slope

A function is a rule that assigns exactly one output to each input. On PSAT 8/9, you’ll mostly work with linear functions, which produce straight-line graphs.

Linear functions matter because they model constant change: each time you increase the input by $1$, the output changes by the same amount.

Function notation and meaning

If a function is named $f$, then $f(x)$ means “the output when the input is $x$.” It is not multiplication.

For example, if:

$f(x) = 2x + 3$

then:

$f(4) = 2(4) + 3 = 11$

A common mistake is reading $f(x)$ as $f \times x$.

The coordinate plane and graphing points

A point $\left(x, y\right)$ means move $x$ units left/right and $y$ units up/down.

• If $x$ is negative, move left.
• If $y$ is negative, move down.

Slope as a rate of change

Slope measures how steep a line is. Conceptually, it is “how much $y$ changes when $x$ changes.”

Between two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

You can think of it as:

$m = \frac{\text{rise}}{\text{run}}$

Slope matters because it often represents a real-world rate, like dollars per hour or miles per minute.

Example (slope from two points)

Find the slope through $\left(2, 5\right)$ and $\left(6, 13\right)$.

$m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$

A common mistake is swapping differences on top but not on bottom. Swapping both is fine; swapping only one changes the sign.

The slope-intercept form

A common linear form is:

$y = mx + b$

• $m$ is the slope
• $b$ is the $y$-intercept (where the line crosses the $y$-axis), i.e., the value of $y$ when $x = 0$

Example (interpret)

In $y = 3x + 4$:

• Slope $3$ means $y$ increases by $3$ for every $1$ increase in $x$.
• Intercept $4$ means when $x = 0$, $y = 4$.

A frequent mistake is treating $b$ as the $x$-intercept.

Writing a linear equation from information

You can build a line from:

1. Slope and a point (often easiest via slope-intercept after finding $b$)
2. Two points (compute slope first)
3. A table or graph (identify rise/run and intercept)

Example (line from slope and point)

A line has slope $m = -2$ and passes through $\left(3, 1\right)$. Find its equation in slope-intercept form.

Start with:

$y = -2x + b$

Plug in the point:

$1 = -2(3) + b$

$1 = -6 + b$

$b = 7$

So:

$y = -2x + 7$

Intercepts and what they mean

• The $y$-intercept occurs when $x = 0$.
• The $x$-intercept occurs when $y = 0$.

To find the $x$-intercept, set $y = 0$ and solve.

Example (find intercepts)

For:

$y = 2x - 6$

$y$-intercept: set $x = 0$:

$y = -6$

So $\left(0, -6\right)$.

$x$-intercept: set $y = 0$:

$0 = 2x - 6$

$2x = 6$

$x = 3$

So $\left(3, 0\right)$.

Linear vs. non-linear graphs

A linear relationship has a constant rate of change (straight line). If the rate changes, the graph curves and the relationship is non-linear.

One quick check from a table: if $x$ increases by equal amounts, and $y$ changes by equal amounts, it’s linear.

Exam Focus

• Typical question patterns:
  • Identify slope from a graph, table, or two points.
  • Interpret slope and intercept in context (rate and starting value).
  • Compare two linear models (which grows faster, which starts higher).
• Common mistakes:
  • Confusing slope with intercept (mixing up $m$ and $b$).
  • Using $\frac{x_2 - x_1}{y_2 - y_1}$ instead of the correct slope formula.
  • Treating $f(x)$ as multiplication.

Systems of Linear Equations and Inequalities

A system is a set of equations or inequalities considered together. The solution must satisfy all conditions at once. Systems are powerful because real situations often have multiple constraints.

Systems of two linear equations

A system of two linear equations can have:

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

Solving by substitution

Substitution works well when one equation is already solved for a variable.

Example (substitution)

Solve:

$y = 2x + 1$

$3x + y = 16$

Substitute $y = 2x + 1$ into the second equation:

$3x + (2x + 1) = 16$

$5x + 1 = 16$

$5x = 15$

$x = 3$

Then:

$y = 2(3) + 1 = 7$

Solution: $\left(3, 7\right)$.

Common mistakes include substituting into the wrong place or forgetting parentheses, especially when substituting an expression with a negative sign.

Solving by elimination

Elimination adds or subtracts equations to eliminate one variable.

Example (elimination)

Solve:

$2x + 3y = 13$

$2x - y = 1$

Subtract the second equation from the first to eliminate $x$:

$(2x + 3y) - (2x - y) = 13 - 1$

$4y = 12$

$y = 3$

Plug back into $2x - y = 1$:

$2x - 3 = 1$

$2x = 4$

$x = 2$

Solution: $\left(2, 3\right)$.

A common mistake is adding equations when you needed to subtract (or vice versa). Always check which operation actually cancels a variable.

Systems in word problems

Many system problems are “two quantities, two conditions.” You define variables and write two equations.

Example (tickets)

A school sold $50$ tickets for a total of $\$310$. Adult tickets cost $\$8$ and student tickets cost $\$5$. How many of each were sold?

Let $a$ be adult tickets and $s$ be student tickets.

Count equation:

$a + s = 50$

Money equation:

$8a + 5s = 310$

Substitute $s = 50 - a$ into the money equation:

$8a + 5(50 - a) = 310$

$8a + 250 - 5a = 310$

$3a = 60$

$a = 20$

Then:

$s = 50 - 20 = 30$

So $20$ adult and $30$ student tickets.

Systems of inequalities

A system of inequalities describes a region of points that satisfy all inequalities. You’ll often see these in contexts like budgets, capacity limits, or “at least/at most” constraints.

Key interpretation ideas:

• $y > 2x + 1$ means points above the line $y = 2x + 1$.
• $y \geq 2x + 1$ includes the boundary line.

On PSAT 8/9, you may be asked to identify which point satisfies a system or to match a shaded graph to inequalities.

Example (test a point)

Does $\left(1, 4\right)$ satisfy:

$y \geq x + 2$

$y < 6$

Check:

$4 \geq 1 + 2$ is true.

$4 < 6$ is true.

So yes, $\left(1, 4\right)$ satisfies the system.

A common mistake is testing the wrong coordinate (plugging $y$ into the $x$ spot).

Exam Focus

• Typical question patterns:
  • Solve a system and interpret the solution as an ordered pair.
  • Determine whether a system has one, none, or infinitely many solutions.
  • Identify a point that satisfies a system of inequalities.
• Common mistakes:
  • Forgetting parentheses during substitution.
  • Making sign errors when adding/subtracting equations.
  • Confusing strict and non-strict inequalities when interpreting boundary lines.

Data, Statistics, and Probability

This section tests whether you can read information, describe it accurately, and make reasonable calculations. The hardest part is often choosing the right measure (mean vs. median) and interpreting what a statistic implies.

Reading tables and graphs

You should be comfortable with:

• Bar graphs and line graphs (trends, comparisons)
• Scatterplots (association, clusters, outliers)
• Two-way tables (counts by two categories)

A key skill is identifying what the axes and labels actually represent before you compute anything. Many wrong answers come from misreading the scale.

Measures of center: mean and median

The mean is the average:

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

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

Why both matter: the mean uses all values but is sensitive to extreme values (outliers). The median resists outliers and better describes the “typical” value when the distribution is skewed.

Example (mean vs. median)

Data: $2, 3, 3, 4, 20$

Mean:

$\frac{2+3+3+4+20}{5} = \frac{32}{5} = 6.4$

Median is the middle value: $3$.

Here the outlier $20$ pulls the mean up. Median better captures the typical small values.

A common mistake is computing the median without ordering the data.

Measures of spread: range and interquartile range

The range is:

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

The interquartile range (IQR) measures the spread of the middle $50\%$ of the data:

$\text{IQR} = Q_3 - Q_1$

Where $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half. IQR is useful because it is not very affected by outliers.

Box plots use median, quartiles, and sometimes outliers to show distribution shape.

Interpreting scatterplots

A scatterplot compares two quantitative variables.

• A positive association means as $x$ increases, $y$ tends to increase.
• A negative association means as $x$ increases, $y$ tends to decrease.
• No association means no clear pattern.

An outlier is a point far from the pattern. Outliers matter because they can distort averages and trend lines.

You might see a line of best fit (informal) used to predict values. Predictions outside the data range are extrapolation and are less reliable.

Example (prediction idea)

If a line of best fit suggests that when $x = 10$, $y \approx 30$, then $30$ is a reasonable prediction if $10$ is within the observed $x$ values.

Common mistakes include assuming correlation means causation. A trend does not prove that changing $x$ causes $y$ to change.

Probability basics

Probability measures how likely an event is:

$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{number of possible outcomes}}$

Probabilities range from $0$ to $1$.

Compound probability (simple cases)

Some PSAT 8/9 questions involve multiple steps, like drawing without replacement or using two-way tables.

Example (two-way table probability)

Suppose a class has $12$ students who play a sport and $8$ who do not. Of the sport players, $5$ also play an instrument. Of the non-sport students, $2$ play an instrument.

Total students:

$12 + 8 = 20$

Total instrument players:

$5 + 2 = 7$

Probability a randomly chosen student plays an instrument:

$\frac{7}{20}$

A common mistake is using the wrong total—always clarify what the “sample space” is for the question.

Exam Focus

• Typical question patterns:
  • Compute mean/median/range from small data sets or from a frequency table.
  • Interpret scatterplots: direction, strength, outliers, and reasonable predictions.
  • Probability from two-way tables and simple scenarios.
• Common mistakes:
  • Forgetting to order data when finding the median or quartiles.
  • Misreading graph scales (especially when intervals are not $1$).
  • Using the wrong denominator in probability (not matching the described sample space).

Geometry and Measurement (Including Coordinate Geometry)

Geometry questions on PSAT 8/9 focus on relationships between shapes, using formulas correctly, and reasoning about lengths and angles. Many errors come from confusing similar formulas (like area vs. perimeter) or mixing units.

Angles, lines, and basic facts

An angle measures rotation. Important relationships:

• Vertical angles (opposite angles formed by two intersecting lines) are equal.
• Supplementary angles add to $180$ degrees.
• Complementary angles add to $90$ degrees.

If two lines are parallel and cut by a transversal, certain angles match (corresponding and alternate interior angles are equal). Problems often ask you to find unknown angle measures using these equalities.

Triangles and the triangle inequality

A triangle has three sides and three angles summing to $180$ degrees.

The triangle inequality says the sum of any two side lengths must be greater than the third. This helps you decide if a triangle is possible.

The Pythagorean theorem

In a right triangle (one $90$ degree angle), with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

This matters constantly in coordinate geometry and distance problems.

Example (find hypotenuse)

Legs $6$ and $8$:

$6^2 + 8^2 = c^2$

$36 + 64 = c^2$

$100 = c^2$

$c = 10$

A common mistake is adding the legs directly (thinking $c = a+b$).

Area and perimeter (what’s being measured)

• Perimeter is distance around a shape (linear units).
• Area is the amount of surface inside (square units).

Common area formulas:

• Rectangle:

$A = lw$

• Triangle:

$A = \frac{1}{2}bh$

• Parallelogram:

$A = bh$

Where $b$ is base and $h$ is the perpendicular height.

Example (triangle area)

A triangle has base $10$ and height $7$:

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

A common mistake is using a slanted side as height. Height must be perpendicular to the base.

Circles: circumference and area

Circle formulas:

• Circumference:

$C = 2\pi r$

• Area:

$A = \pi r^2$

Where $r$ is radius. Diameter $d$ is $2r$.

Example (circle area)

If $r = 5$:

$A = \pi(5^2) = 25\pi$

A common mistake is using $\pi r$ for circumference or forgetting to square the radius for area.

Volume of rectangular prisms

Volume is “how much space inside,” measured in cubic units.

For a rectangular prism:

$V = lwh$

Example (volume)

A box $4$ by $3$ by $10$ has volume:

$V = 4 \times 3 \times 10 = 120$

Distance in the coordinate plane

The distance between $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ comes from the Pythagorean theorem:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Example (distance)

Distance between $\left(1, 2\right)$ and $\left(5, 5\right)$:

$d = \sqrt{(5-1)^2 + (5-2)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

A common mistake is forgetting to square the differences or squaring coordinates separately.

Slope and parallel/perpendicular lines

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals (when both slopes are defined).

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

$-\frac{1}{m}$

Example (perpendicular slope)

If $m = 3$, perpendicular slope is:

$-\frac{1}{3}$

If the slope is $0$ (horizontal line), the perpendicular line is vertical (undefined slope).

Exam Focus

• Typical question patterns:
  • Use area/volume formulas in word problems with unit attention.
  • Apply the Pythagorean theorem and coordinate distance.
  • Angle-finding with parallel lines or triangle angle sums.
• Common mistakes:
  • Confusing area and perimeter, or using inconsistent units.
  • Using the wrong “height” in area formulas (not perpendicular).
  • Forgetting to square the radius in $A = \pi r^2$.

Nonlinear Relationships: Exponents and Quadratics (Intro Level)

PSAT 8/9 includes some “next step” algebra ideas—especially exponent growth and quadratic expressions. You are usually not expected to do advanced solving methods, but you should recognize forms and reason about them.

Exponential growth and repeated multiplication

An exponential expression grows by a constant factor, not a constant amount. A common form is:

$a \times b^t$

• $a$ is the starting value (when $t = 0$)
• $b$ is the growth factor per time step
• $t$ counts steps (often years, days, or cycles)

Example (growth factor)

A population starts at $200$ and increases by $10\%$ per year. The growth factor is:

$1 + 0.10 = 1.1$

After $t$ years:

$200 \times 1.1^t$

A common mistake is adding $10\%$ each year as a fixed amount rather than multiplying by $1.1$ each year.

Quadratic expressions and what makes them special

A quadratic expression includes a squared variable term, like:

$x^2$

A quadratic function often looks like:

$y = ax^2 + bx + c$

Its graph is a parabola, not a line. On PSAT 8/9, the most common tasks are simplifying quadratics, factoring simple ones, and solving by factoring when it’s straightforward.

Multiplying binomials (building a quadratic)

A binomial has two terms, like $x+3$. Multiplying binomials is a structured use of distribution.

Example (multiply)

Multiply:

$(x + 4)(x + 2)$

Distribute each term in the first binomial across the second:

$x(x+2) + 4(x+2)$

$= x^2 + 2x + 4x + 8$

$= x^2 + 6x + 8$

A common mistake is forgetting one of the middle products.

Factoring simple quadratics

Factoring is the reverse: write a quadratic as a product.

For:

$x^2 + bx + c$

You look for two numbers that multiply to $c$ and add to $b$.

Example (factor)

Factor:

$x^2 + 7x + 12$

We need two numbers that multiply to $12$ and add to $7$: $3$ and $4$.

So:

$x^2 + 7x + 12 = (x+3)(x+4)$

A common mistake is choosing numbers that multiply correctly but add to the wrong value.

Solving quadratics by factoring (when set to zero)

If:

$(x+p)(x+q) = 0$

then either factor can be zero:

$x+p = 0 \quad \text{or} \quad x+q = 0$

This is the zero product property.

Example (solve by factoring)

Solve:

$x^2 + 7x + 12 = 0$

Factor:

$(x+3)(x+4) = 0$

Set each factor to zero:

$x+3 = 0 \Rightarrow x = -3$

$x+4 = 0 \Rightarrow x = -4$

So the solutions are $-3$ and $-4$.

A common mistake is trying to “cancel” an $x$ across terms or dividing by $x$ (which can lose solutions).

Exam Focus

• Typical question patterns:
  • Identify whether a relationship is linear or exponential from a description or table.
  • Multiply or factor simple quadratics with integer coefficients.
  • Solve a quadratic equation by factoring when it factors cleanly.
• Common mistakes:
  • Treating exponential growth as repeated addition instead of repeated multiplication.
  • Factoring errors: wrong pair for $b$ and $c$.
  • Forgetting that a quadratic can have two solutions (two roots).