PSAT 10 Advanced Math Review: Quadratics, Exponentials & Rational/Radical Equations

What You Need to Know (High-Yield Map)

This is the “Advanced Math” core: you’ll be asked to rewrite, solve, and interpret equations involving:

• Quadratics (parabolas): factoring, vertex/roots, max/min, equation forms.
• Exponentials: growth/decay, rewrite with exponent rules, compare values.
• Rational equations (fractions with variables) and radical equations (square roots): solve while watching domain restrictions and extraneous solutions.

Why it matters: PSAT questions often reward the student who can quickly choose the best form and avoid traps (especially extraneous solutions and sign errors).

Critical reminder: When you solve rational or radical equations, you must check your answers in the original equation. Squaring and clearing denominators can create fake solutions.

Step-by-Step Breakdown

A) Quadratics: choose the fastest path

1) If it’s set equal to $0$, try factoring first

1. Rewrite into standard form:

$ax^2+bx+c=0$

2. Factor (if possible) into:

$(px+q)(rx+s)=0$

3. Use the zero product property:

$px+q=0 \quad \text{or} \quad rx+s=0$

4. Solve each linear equation.

Mini-example

$x^2-5x+6=0$

Factor:

$(x-2)(x-3)=0$

So:

$x=2 \quad \text{or} \quad x=3$

2) If it doesn’t factor nicely, use the quadratic formula

1. Identify $a$, $b$, $c$ from:

$ax^2+bx+c=0$

2. Plug into:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

3. Simplify; if asked about number of solutions, check the discriminant:

$\Delta=b^2-4ac$

3) If you need the vertex (max/min), use vertex form or completing the square

Vertex form:

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

Vertex is:

$(h,\,k)$

To complete the square for:

$y=ax^2+bx+c$

1. Factor out $a$ from the $x$-terms (if $a\neq 1$).
2. Take half of the coefficient of $x$, square it, add and subtract it.
3. Rewrite as a square plus/minus a constant.

Mini-example (vertex quickly)

$y=x^2-6x+5$

Complete the square:

$y=(x-3)^2-4$

Vertex:

$(3,\,-4)$

Maximum/minimum value is:

$-4 \text{ (minimum, since } a>0\text{)}$

B) Exponentials: rewrite to compare/solve

1) Use exponent rules to rewrite into common bases

Key goal: turn both sides into the same base.

Example approach:

$2^{x+1}=16$

Rewrite $16$ as a power of $2$:

$16=2^4$

So:

$2^{x+1}=2^4 \Rightarrow x+1=4 \Rightarrow x=3$

2) Growth/decay: identify the multiplier

Exponential model:

$A(t)=A_0\,b^t$

• Growth if $b>1$
• Decay if $0<b<1$

Percent form:

$A(t)=A_0(1+r)^t$

• Growth rate $r$ if $r>0$
• Decay rate $r$ if $r<0$

Decision point: If the problem says “increases by $p\%$ each period,” then:

$b=1+\frac{p}{100}$

If it says “decreases by $p\%$ each period,” then:

$b=1-\frac{p}{100}$

C) Rational equations: restrict, clear denominators, check

Given an equation with fractions (like $\frac{1}{x-2}$), do this:

1. State restrictions: denominators cannot be $0$.

Example restriction for $\frac{1}{x-2}$:

$x\neq 2$

2. Find the LCD (least common denominator).
3. Multiply every term by the LCD to clear denominators.
4. Solve the resulting equation.
5. Check solutions in the original and reject any that violate restrictions.

Mini-example

$\frac{2}{x}+1=\frac{5}{x}$

Restriction:

$x\neq 0$

Multiply by $x$:

$2+x=5$

So:

$x=3$

Check: works.

D) Radical equations: isolate the radical, square, check

1. Isolate the radical term.
2. Square both sides.
3. Solve.
4. Check in the original (squaring can create extraneous solutions).

Mini-example

$\sqrt{x+5}=x-1$

Domain clue: right side must be nonnegative:

$x-1\ge 0 \Rightarrow x\ge 1$

Square:

$x+5=(x-1)^2$

Expand:

$x+5=x^2-2x+1$

Rearrange:

$0=x^2-3x-4$

Factor:

$0=(x-4)(x+1)$

Candidates:

$x=4 \quad \text{or} \quad x=-1$

Check with $x\ge 1$, and plug in: only

$x=4$

works.

Key Formulas, Rules & Facts

Quadratics (must-know)

Exponent rules (use constantly)

Exponential models

Rational & radical equation rules

Examples & Applications

Example 1 (Quadratic: interpret from form)

Given:

$y=2(x-3)^2+5$

• Vertex:

$(3,\,5)$

• Opens upward since:

$a=2>0$

• Minimum value is:

$5$

• Axis of symmetry:

$x=3$

PSAT angle: questions love “minimum value,” “vertex,” and “where is it symmetric.”

Example 2 (Quadratic: discriminant without fully solving)

How many real solutions does

$3x^2+2x+10=0$

have?

Compute:

$\Delta=b^2-4ac=(2)^2-4(3)(10)=4-120=-116$

Since:

$\Delta<0$

there are no real solutions.

Example 3 (Exponential: growth/decay)

A population starts at

$200$

and increases by

$15\%$

per year. Expression after $t$ years:

$P(t)=200(1.15)^t$

If it instead decreases by $15\%$ per year:

$P(t)=200(0.85)^t$

PSAT angle: identify the multiplier correctly.

Example 4 (Rational/Radical: solve + extraneous check)

Solve:

$\frac{x}{x-2}=3$

Restriction:

$x\neq 2$

Multiply both sides by $x-2$:

$x=3(x-2)$

Solve:

$x=3x-6 \Rightarrow -2x=-6 \Rightarrow x=3$

Check restriction: $3\neq 2$, so solution is:

$x=3$

Now a radical trap:

$\sqrt{x}=x-2$

Domain clue:

$x\ge 0 \quad \text{and} \quad x-2\ge 0 \Rightarrow x\ge 2$

Square:

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

Rearrange:

$0=x^2-5x+4$

Factor:

$0=(x-1)(x-4)$

Candidates:

$x=1 \quad \text{or} \quad x=4$

Domain requires $x\ge 2$, so only $x=4$ remains. Check: works.

Common Mistakes & Traps

1. Forgetting denominator restrictions

   • Wrong move: solving a rational equation and accepting a value that makes a denominator $0$.
   • Why wrong: the original expression is undefined there.
   • Fix: write restrictions first (e.g., $x\neq 2$).

2. Not checking after clearing fractions

   • Wrong move: multiplying by an expression containing $x$ and forgetting it might be $0$.
   • Why wrong: the algebra step can hide invalid solutions.
   • Fix: always plug final answers back into the original rational equation.

3. Not checking after squaring (extraneous solutions)

   • Wrong move: squaring both sides in a radical equation and keeping all solutions.
   • Why wrong: squaring is not a reversible operation; it can turn a false statement into a true one.
   • Fix: check each candidate in the original equation.

4. Sign mistakes with the quadratic formula

   • Wrong move: using $b$ instead of $-b$ or misplacing parentheses.
   • Why wrong: small sign errors completely change the roots.
   • Fix: write it exactly as:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

5. Mixing up vertex form signs

   • Wrong move: thinking $y=a(x-3)^2+k$ has vertex $(-3,k)$.
   • Why wrong: vertex is $\bigl(h,k\bigr)$ where the form is $a(x-h)^2+k$.
   • Fix: remember it’s “opposite sign inside.”

6. Exponent rule misuse across addition

   • Wrong move: simplifying $(a+b)^2$ as $a^2+b^2$.
   • Why wrong: exponent distribution only works for multiplication, not addition.
   • Fix: expand correctly:

$(a+b)^2=a^2+2ab+b^2$

7. Dropping parentheses with negative exponents

   • Wrong move: treating $\left(2x\right)^{-1}$ as $2x^{-1}$.
   • Why wrong: $\left(2x\right)^{-1}=\frac{1}{2x}$, but $2x^{-1}=\frac{2}{x}$.
   • Fix: keep bases grouped.

8. Assuming a square root can be negative

   • Wrong move: stating $\sqrt{9}=-3$.
   • Why wrong: the principal square root is nonnegative.
   • Fix:

$\sqrt{9}=3$

Memory Aids & Quick Tricks

Quick Review Checklist

• Quadratics:
  • Can you move everything to

$ax^2+bx+c=0$

• Can you factor quickly when possible?
• Do you remember:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

• Can you identify vertex from

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

• Can you use:

$\Delta=b^2-4ac$

to count real solutions?

• Exponentials:
  • Do you apply exponent rules only when bases match?
  • Can you turn percent change into multiplier:

$1\pm \frac{p}{100}$

• Can you rewrite numbers as powers of a common base (like $8=2^3$, $16=2^4$, $27=3^3$)?

  • Rational equations:

• Did you list restrictions (denominators $\neq 0$) before solving?

• Did you multiply by the LCD and then check your solution?

  • Radical equations:

• Did you isolate the radical before squaring?

• Did you check for extraneous solutions after squaring?

You’ve got this: be disciplined about form, domain, and checking, and these problems become very predictable.