Intermediate Algebra

Quadratics, functions, polynomials, rational expressions, logs and sequences.

📐 Quadratic Equations

Part 1 of 7 — Factoring, the Quadratic Formula & the Discriminant

Quadratics come up often on ACT Math, in both the Algebra and Functions questions. A quadratic equation has the standard form:

$ax^2 + bx + c = 0$

Three core solving techniques:

Quadratic Formula:

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

The expression under the radical, $\Delta = b^2 - 4ac$, is the discriminant and tells you how many real solutions exist.

Sum & product shortcut: For $ax^2 + bx + c = 0$, the two solutions add to $-\frac{b}{a}$ and multiply to $\frac{c}{a}$ — no solving needed. Example: for $x^2 - 5x + 6 = 0$ (roots $2$ and $3$), the sum is $-\frac{-5}{1} = 5$ and the product is $\frac{6}{1} = 6$.

🔧 Functions & Notation

Part 2 of 7 — Evaluating f(x), Domain & Range, Composition

A function is a rule that assigns exactly one output to each input. On the ACT you'll see standard notation like $f(x)$, $g(x)$, etc.

Evaluating a function means substituting a value for $x$.

Example 1: If $f(x) = 3x^2 - 2x + 1$, find $f(4)$.

$f(4) = 3(16) - 2(4) + 1 = 48 - 8 + 1 = 41$

Example 2: If $g(x) = \frac{x+1}{x-3}$, find $g(5)$.

$g(5) = \frac{5+1}{5-3} = \frac{6}{2} = 3$

📊 Polynomials

Part 3 of 7 — Operations, Factoring, Zeros & the Remainder Theorem

A polynomial in $x$ is an expression like:

$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$

The degree is the highest power of $x$ with a nonzero coefficient.

Key fact: A polynomial of degree $n$ has at most $n$ real zeros.

🔢 Rational Expressions

Part 4 of 7 — Simplifying, Adding/Subtracting & Complex Fractions

A rational expression is a fraction whose numerator and denominator are polynomials:

$\frac{P(x)}{Q(x)}, \quad Q(x) \neq 0$

Simplifying means cancelling common factors from top and bottom.

Example 1: Simplify $\frac{x^2 - 9}{x + 3}$.

$\frac{(x-3)(x+3)}{x+3} = x - 3, \quad x \neq -3$

Example 2: Simplify $\frac{2x^2 + 6x}{4x}$.

$\frac{2x(x + 3)}{4x} = \frac{x + 3}{2}, \quad x \neq 0$

Always state the restriction(s) on $x$!

📈 Logarithms & Exponents

Part 5 of 7 — Log Rules, Solving Log Equations & Change of Base

The logarithm $\log_b a = c$ means $b^c = a$.

Key log rules:

Also remember: $\log_b b = 1$ and $\log_b 1 = 0$ for any valid base $b$.

🔗 Sequences & Series

Part 6 of 7 — Arithmetic, Geometric, nth Term & Partial Sums

A sequence is an ordered list of numbers. A series is the sum of a sequence's terms.

Arithmetic sequence: $a_n = a_1 + (n-1)d$

Geometric sequence: $a_n = a_1 \cdot r^{n-1}$

Example 1 — Arithmetic: $3, 7, 11, 15, \ldots$ Here $a_1 = 3$, $d = 4$.

$a_{10} = 3 + (10-1)(4) = 3 + 36 = 39$

Example 2 — Geometric: $2, 6, 18, 54, \ldots$ Here $a_1 = 2$, $r = 3$.

$a_5 = 2 \cdot 3^{4} = 2 \cdot 81 = 162$

🏆 Review & Mixed Practice

Part 7 of 7 — Cheat Sheet & Mixed ACT Intermediate Algebra Problems

Here's a quick-reference sheet covering every major topic from Parts 1–6.

Cheat Sheet