PSAT 8/9 Advanced Math: Expressions, Nonlinear Equations, and Nonlinear Functions

Equivalent expressions

An expression is a mathematical phrase with numbers, variables, and operations (for example, $3x+7$ or $x^2-4$). Two expressions are **equivalent** if they have the **same value for every allowed value of the variable(s)**. For instance, $2(x+5)$ and $2x+10$ are equivalent because they always give the same result no matter what number you substitute for $x$.

What “equivalent” really means (and why it matters)

Equivalence is about preserving meaning while changing form. In math, changing form is powerful because different forms make different tasks easier:

• If you want to plug in values quickly, an expanded form like $2x+10$ may be convenient.
• If you want to solve an equation, a factored form like $(x-3)(x+3)$ can reveal solutions immediately.
• If you want to see structure (like a repeated factor), a rewritten form can help you notice patterns.

On the PSAT 8/9, you’re often asked to rewrite expressions, choose an equivalent expression from answer choices, or recognize which transformation keeps an expression the same.

A key warning: equivalence depends on the domain—the values you’re allowed to substitute. For example, $\frac{x^2-1}{x-1}$ simplifies to $x+1$, but only when $x \ne 1$ because division by zero is not allowed. The simplified expression matches the original everywhere the original is defined.

Core tools for creating equivalent expressions

Distributing and factoring are opposite moves

The distributive property lets you multiply a factor across terms:

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

Factoring reverses distributing—pulling out a common factor.

Why this matters: distributing helps you combine like terms; factoring helps you see zeros and solve equations.

Example 1: Distribute then simplify
Rewrite $3(2x-5)+4x$ as an equivalent expression.

1. Distribute $3$:

   $3(2x-5)=6x-15$

2. Add $4x$ and combine like terms:

   $6x-15+4x=10x-15$

So an equivalent expression is:

$10x-15$

Common slip: forgetting to distribute to both terms, especially the negative one (turning $3(2x-5)$ into $6x-5$ is incorrect).

Example 2: Factor out the greatest common factor (GCF)
Factor $12x+18$.

1. The GCF of $12$ and $18$ is $6$.
2. Factor out $6$:

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

Both forms are equivalent; the factored form shows the shared factor clearly.

Combining like terms

Like terms have the same variable part (same variables raised to the same powers). For example, $7x$ and $-2x$ are like terms; $7x$ and $7x^2$ are not.

Example:

$5x^2+3x-2x^2+10x = 3x^2+13x$

A frequent mistake is combining terms that “look similar” but aren’t like terms, such as trying to add $x$ and $x^2$.

Exponent rules (with meaning, not memorization)

Exponents represent repeated multiplication. That’s why these rules work.

If $a$ is a nonzero number and $m$ and $n$ are integers:

• Multiply same base:

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

  You’re combining repeated multiplication of the same base.

• Divide same base:

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

  You’re canceling factors.

• Power of a power:

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

• Power of a product:

  $(ab)^n=a^n b^n$

• Zero exponent (for nonzero base):

  $a^0=1$

Example 3: Simplify using exponent rules
Simplify:

$\frac{(x^3)^2 \cdot x}{x^4}$

1. Power of a power:

$(x^3)^2=x^6$

2. Multiply same base in the numerator:

$x^6 \cdot x=x^7$

3. Divide same base:

$\frac{x^7}{x^4}=x^{7-4}=x^3$

So the simplified expression is:

$x^3$

Important restriction: This simplification assumes $x \ne 0$ because you cannot divide by zero in $\frac{x^7}{x^4}$.

Common algebraic identities (pattern-based equivalence)

Some expressions appear so often that recognizing their patterns saves time.

Memory aid: “Square–square gives sum and difference” for $a^2-b^2$.

Example 4: Factor using a pattern
Factor:

$x^2-16$

This is a difference of squares because $16=4^2$:

$x^2-4^2=(x-4)(x+4)$

A very common mistake is trying to factor $x^2+16$ the same way—over real numbers, $x^2+16$ does not factor into real linear factors.

Rational expressions: simplifying without “illegal moves”

A rational expression is a fraction with expressions in the numerator and/or denominator, like $\frac{x^2-1}{x-1}$. You simplify by factoring and canceling common factors—not by canceling terms that are separated by addition.

Example 5: Simplify a rational expression

$\frac{x^2-1}{x-1}$

1. Factor the numerator (difference of squares):

$x^2-1=(x-1)(x+1)$

2. Cancel the common factor $x-1$:

$\frac{(x-1)(x+1)}{x-1}=x+1$

But you must state the restriction:

$x \ne 1$

because the original expression is undefined at $x=1$.

A classic error is canceling across addition, such as claiming:

$\frac{x+3}{x+5}=\frac{3}{5}$

That is not valid because $x+3$ and $x+5$ are not factors you can cancel.

Notation reference: radicals and fractional exponents

Sometimes the same idea is written in two forms.

Be careful with $\sqrt{x^2}$: it equals $|x|$, not always $x$, because square roots are defined as nonnegative.

Exam Focus

• Typical question patterns:
  • Choose which answer choice is equivalent to a given expression after expanding or factoring.
  • Simplify expressions with exponents (including products/quotients of powers).
  • Simplify rational expressions by factoring and canceling, sometimes asking for restrictions like $x \ne 2$.
• Common mistakes:
  • Distributing incorrectly with negatives (missing a sign change).
  • Canceling terms instead of factors in rational expressions.
  • Using exponent rules across addition, such as thinking $(a+b)^2=a^2+b^2$ (it is not).

Nonlinear equations in one variable and systems of equations in two variables

A nonlinear equation is an equation where the variable is not just to the first power in a simple way. If you see things like $x^2$, $|x|$, $\sqrt{x}$, or a variable multiplied by itself, you’re usually dealing with nonlinear behavior.

What makes nonlinear equations different (and why that matters)

Linear equations (like $2x+3=11$) generally have one solution and graphs that are straight lines. Nonlinear equations can have two solutions, one solution, or no real solutions, depending on the equation. That matters on the PSAT 8/9 because you must be ready for:

• More than one valid answer (especially with quadratics).
• Checking solutions to avoid extraneous answers (especially when squaring both sides).
• Systems where substituting one equation into another produces a quadratic.

Solving nonlinear equations in one variable

Quadratic equations: the “shape” of two solutions

A quadratic equation has the form:

$ax^2+bx+c=0$

where $a$, $b$, and $c$ are numbers and $a \ne 0$.

You typically solve quadratics on the PSAT 8/9 using methods that rely on structure:

1. Factoring (when it factors nicely)
2. Square root method (when it’s in the form $x^2=k$ or $(x-h)^2=k$)

Method 1: Solving by factoring

If you can rewrite a quadratic as a product equal to zero, you can use the zero product property:

If $pq=0$, then $p=0$ or $q=0$.

Example 1: Solve by factoring
Solve:

$x^2-5x=0$

1. Factor out the GCF $x$:

$x(x-5)=0$

2. Use the zero product property:

$x=0 \quad \text{or} \quad x-5=0$

3. Solve the second equation:

$x=5$

So the solutions are:

$x=0 \quad \text{and} \quad x=5$

A common mistake is to “divide both sides by $x$” immediately. That can accidentally throw away the solution $x=0$.

Method 2: Square root method

This is best when the variable is squared and isolated.

Example 2: Solve by taking square roots
Solve:

$(x-2)^2=25$

1. Take the square root of both sides, remembering both the positive and negative roots:

$x-2=5 \quad \text{or} \quad x-2=-5$

2. Solve each:

$x=7 \quad \text{or} \quad x=-3$

So the solutions are:

$x=7 \quad \text{and} \quad x=-3$

A frequent error is taking only the positive root and missing $x=-3$.

Equations with radicals: isolate, then square carefully

Sometimes you’ll see equations like:

$\sqrt{x+1}=3$

To solve, you isolate the radical and then square both sides. Squaring can introduce solutions that don’t actually work, so you should check your solution in the original equation.

Example 3: Solve a radical equation
Solve:

$\sqrt{x+1}=3$

1. Square both sides:

$x+1=9$

2. Solve:

$x=8$

3. Check in the original:

$\sqrt{8+1}=\sqrt{9}=3$

It checks, so:

$x=8$

Systems involving nonlinear equations (two variables)

A system of equations asks for values of $x$ and $y$ that make both equations true at the same time. When one equation is nonlinear (often a quadratic), the system can have 0, 1, or 2 solutions.

How systems connect to graphs

Each equation represents a graph.

• A linear equation graphs as a line.
• A quadratic function graphs as a parabola.

A solution to the system is an intersection point—a point where the graphs cross.

Solving a linear–quadratic system by substitution

Substitution is common because one equation is often already solved for $y$.

Example 4: Solve a system (line and parabola)
Solve:

$y=x^2$

$y=2x+3$

1. Since both equal $y$, set them equal to each other:

$x^2=2x+3$

2. Put everything on one side:

$x^2-2x-3=0$

3. Factor:

$(x-3)(x+1)=0$

4. Solve for $x$:

$x=3 \quad \text{or} \quad x=-1$

5. Plug into either original equation to find $y$.

If $x=3$:

$y=3^2=9$

If $x=-1$:

$y=(-1)^2=1$

So the solutions are the ordered pairs:

$(3,9) \quad \text{and} \quad (-1,1)$

What can go wrong:

• Forgetting to find $y$ after finding $x$.
• Making a sign error when moving terms, turning $x^2-2x-3$ into $x^2-2x+3$.

When a system has one or no solutions

If a line is tangent to a parabola, the system has one intersection point. If a line never intersects the parabola, there are no real solutions. On the PSAT 8/9, this can be tested conceptually by asking how many solutions a system has based on a graph.

Exam Focus

• Typical question patterns:
  • Solve a quadratic equation by factoring or by taking square roots.
  • Solve a system where you substitute a linear expression into a quadratic and then solve the resulting quadratic.
  • Determine the number of solutions from a graph (0, 1, or 2 intersections).
• Common mistakes:
  • Missing a solution because you forget the $\pm$ when taking square roots.
  • Dividing by an expression that could be $0$ (losing solutions).
  • Not checking after squaring both sides in radical-type equations, leading to extraneous solutions.

Nonlinear functions

A function is a rule that assigns each input exactly one output. You usually see functions written as $f(x)$, which means “the output of function $f$ when the input is $x$.” A function is nonlinear if its graph is not a straight line—equivalently, its rate of change is not constant.

Why nonlinear functions matter

Linear functions model situations with a constant change per step (like earning the same amount each hour). Many real situations don’t work that way:

• The area of a square grows with $s^2$, not with $s$.
• A ball’s height over time can follow a quadratic relationship.
• Some growth processes multiply repeatedly rather than add repeatedly.

On the PSAT 8/9, nonlinear functions often appear as quadratic functions and may also appear in other forms (like a square root relationship). You’ll be asked to interpret equations, compare outputs, identify features from graphs/tables, and connect forms of an expression to what the graph does.

Function notation and evaluating nonlinear functions

When you see $f(x)=x^2-4x+1$, this is an instruction: replace $x$ with the input value.

Example 1: Evaluate a quadratic function
Given:

$f(x)=x^2-4x+1$

Find $f(3)$.

Substitute $3$ for $x$:

$f(3)=3^2-4(3)+1=9-12+1=-2$

So:

$f(3)=-2$

Common mistake: forgetting parentheses when substituting a negative input, for example using $-2^2$ instead of $(-2)^2$.

Quadratic functions and parabolas

A quadratic function is typically written as:

$f(x)=ax^2+bx+c$

where $a \ne 0$.

What the coefficients tell you

• $a$ controls whether the parabola opens up or down.
  • If $a>0$, it opens upward (has a minimum).
  • If $a<0$, it opens downward (has a maximum).
• $c$ is the $y$-intercept because $f(0)=c$.

The graph is a parabola. Parabolas are symmetric: there is a vertical line through the vertex (the “turning point”) called the axis of symmetry.

Intercepts as “meaningful outputs”

• $y$-intercept: set $x=0$.
• $x$-intercepts (zeros): solve $f(x)=0$.

This is where equivalent expressions connect directly to functions: factoring a quadratic often makes the zeros obvious.

Example 2: Zeros from factored form
Suppose:

$f(x)=(x-2)(x+5)$

To find where the graph crosses the $x$-axis, set $f(x)=0$:

$(x-2)(x+5)=0$

So:

$x=2 \quad \text{or} \quad x=-5$

Those are the $x$-intercepts:

$(2,0) \quad \text{and} \quad (-5,0)$

If the same function were written expanded as $x^2+3x-10$, the intercepts would be much harder to see. This is exactly why equivalence matters: same function, different visibility of features.

Vertex form and “completing the square” (conceptually)

Another useful form is vertex form:

$f(x)=a(x-h)^2+k$

Here, the vertex is $(h,k)$.

You might not always be asked to fully rewrite into vertex form on PSAT 8/9, but you can still use the idea: the expression $(x-h)^2$ is always nonnegative, so the smallest (or largest, if $a<0$) value of the function happens when $(x-h)^2=0$, meaning $x=h$.

Example 3: Identify vertex from vertex form
Given:

$f(x)=(x-4)^2-9$

• Vertex is $(4,-9)$.
• The parabola opens upward (implied $a=1$).
• Minimum value is $-9$.

A typical mistake is reading the vertex as $(-4,-9)$ because of the minus sign. Remember: $x-h$ means the shift is right by $h$.

Other common nonlinear function forms

Square root functions

A basic square root function looks like:

$f(x)=\sqrt{x}$

This is nonlinear because the rate of change decreases as $x$ increases. Domain matters a lot: in real numbers, you need the expression under the root to be nonnegative.

Example 4: Domain reasoning
For:

$g(x)=\sqrt{x-3}$

You need:

$x-3 \ge 0$

so:

$x \ge 3$

So the domain (real inputs allowed) is:

$x \ge 3$

Common mistake: treating the domain like all real numbers without checking the radicand.

Exponential patterns (growth/decay intuition)

Some PSAT 8/9 problems hint at repeated multiplication (for example, “doubles each time”). That’s the signature of an exponential relationship, often written like:

$f(x)=a \cdot b^x$

where $a$ is the starting value and $b$ is the growth factor.

Even if the test doesn’t require advanced solving with logarithms, you should understand the difference in behavior: linear adds the same amount each step; exponential multiplies by the same factor each step.

Example 5: Recognize multiplicative change
If a population starts at $50$ and triples each week, a simple model is:

$P(w)=50 \cdot 3^w$

Then:

$P(2)=50 \cdot 3^2=50 \cdot 9=450$

Connecting representations: equation, table, and graph

Nonlinear functions are often tested by asking you to match or interpret different representations.

• In a table, a linear function has constant first differences (the change in $y$ is constant as $x$ increases by 1). A quadratic function has constant second differences.
• In a graph, linear is a line; quadratic is a parabola; square root has an endpoint and curves.

You don’t need to compute second differences all the time, but knowing the idea helps you recognize a quadratic pattern.

Exam Focus

• Typical question patterns:
  • Evaluate a nonlinear function at a value, including function notation like $f(2)$.
  • Identify features of a quadratic graph (opening direction, intercepts, vertex from a given form or graph).
  • Use a factored form to find zeros or interpret intercepts.
• Common mistakes:
  • Substitution errors with negatives (missing parentheses).
  • Confusing the vertex in $(x-h)^2+k$ as $(-h,k)$ instead of $(h,k)$.
  • Ignoring domain restrictions for square roots (allowing inputs that make the radicand negative).