Intermediate Algebra

📐 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 $\frac{6}{1} = 6$.

Factoring — Worked Examples

Example 1: Solve $x^2 - 5x + 6 = 0$.

We need two numbers whose product is $6$ and sum is $-5$: that's $-2$ and $-3$.

$(x - 2)(x - 3) = 0 \implies x = 2 \text{ or } x = 3$

Example 2: Solve $2x^2 + 7x + 3 = 0$.

We look for factors of $2 \cdot 3 = 6$ that add to $7$: that's $1$ and $6$.

$2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3) = 0$

$x = -\frac{1}{2} \text{ or } x = -3$

Example 3: Using the quadratic formula on $x^2 + 4x - 21 = 0$:

$x = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2} = \frac{-4 \pm 10}{2}$

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

Solving Quadratics 🎯

The Discriminant

The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots:

Example 4: How many real solutions does $x^2 + 6x + 9 = 0$ have?

$\Delta = 6^2 - 4(1)(9) = 36 - 36 = 0$

One repeated root: $x = -3$.

Example 5: How many real solutions does $2x^2 + x + 5 = 0$ have?

$\Delta = 1 - 40 = -39 < 0$

No real solutions.

Quadratic Practice 🧮

1. Solve $x^2 - 7x + 12 = 0$. Enter the smaller root.

2. What is the discriminant of $x^2 + 2x + 5 = 0$?

3. Solve $x^2 - 16 = 0$. Enter the positive root.

Concept Check 🔍

ACT-Style Questions 📋

🔧 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 $g(5)$.

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

Domain & Range

The domain is the set of all valid inputs. The range is the set of all possible outputs.

Common domain restrictions:

• Fractions: denominator $\neq 0$.
• Square roots: radicand $\geq 0$ (for real numbers).
• Logarithms: argument $> 0$.

Example 3: Find the domain of $h(x) = \frac{1}{x^2 - 9}$.

Set $x^2 - 9 \neq 0$: $x \neq \pm 3$. Domain: all real numbers except $3$ and $-3$.

Example 4: Find the domain of $k(x) = \sqrt{2x - 8}$.

$2x - 8 \geq 0 \implies x \geq 4$

Domain: $[4, \infty)$.

Function Evaluation 🎯

Function Composition

The composition $(f \circ g)(x) = f(g(x))$ means plug $g(x)$ into $f$.

Example 5: Let $f(x) = 2x + 3$ and $g(x) = x^2$. Find $(f \circ g)(4)$.

$g(4) = 16 \implies f(16) = 2(16) + 3 = 35$

Example 6: Same functions. Find $(g \circ f)(4)$.

$f(4) = 11 \implies g(11) = 121$

Order matters! $f \circ g \neq g \circ f$ in general.

Example 7: If $f(x) = x + 5$ and $g(x) = 3x$, find $f(g(x))$.

$f(g(x)) = f(3x) = 3x + 5$

Function Practice 🧮

Let $f(x) = 2x - 1$ and $g(x) = x^2 + 3$.

1. What is $f(5)$?

2. What is $g(-2)$?

3. What is $f(g(1))$?

Domain & Composition Concepts 🔍

ACT-Style Questions 📋

📊 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.

Polynomial Operations

Adding/Subtracting: Combine like terms.

$(3x^2 + 2x - 1) + (x^2 - 5x + 4) = 4x^2 - 3x + 3$

Multiplying: Distribute (FOIL for binomials).

$(2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12$

Example — Expand: $(x + 2)^3$

$(x + 2)^3 = x^3 + 3(x^2)(2) + 3(x)(4) + 8 = x^3 + 6x^2 + 12x + 8$

Factoring a cubic: $x^3 - 27 = (x - 3)(x^2 + 3x + 9)$ (difference of cubes).

Polynomial Operations 🎯

Zeros & the Remainder Theorem

A zero (or root) of $P(x)$ is a value $c$ such that $P(c) = 0$.

Factor Theorem: $c$ is a zero of $P(x)$ if and only if $(x - c)$ is a factor.

Remainder Theorem: When $P(x)$ is divided by $(x - c)$, the remainder is $P(c)$.

Example: Let $P(x) = x^3 - 4x^2 + x + 6$. Find $P(2)$.

$P(2) = 8 - 16 + 2 + 6 = 0$

Since $P(2) = 0$, $(x - 2)$ is a factor. Dividing:

$x^3 - 4x^2 + x + 6 = (x - 2)(x^2 - 2x - 3) = (x - 2)(x - 3)(x + 1)$

Zeros: $x = 2, 3, -1$.

Polynomial Practice 🧮

1. What is the degree of $7x^4 + 2x^2 - x + 9$?

2. If $P(x) = x^2 - 5x + 6$, what is $P(3)$?

3. Expand: $(x + 1)(x - 1) = x^2 - \;?$ (enter the constant)

Polynomial Concepts 🔍

ACT-Style Questions 📋

🔢 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 $x$!

Adding & Subtracting

To add or subtract rational expressions, find a common denominator.

Example 3: $\frac{2}{x} + \frac{3}{x+1}$

$= \frac{2(x+1)}{x(x+1)} + \frac{3x}{x(x+1)} = \frac{2x + 2 + 3x}{x(x+1)} = \frac{5x + 2}{x(x+1)}$

Example 4: $\frac{1}{x-2} - \frac{1}{x+2}$

$= \frac{(x+2) - (x-2)}{(x-2)(x+2)} = \frac{4}{x^2 - 4}$

Key tip: Always factor denominators first to spot common factors and find the LCD efficiently.

Simplifying Rationals 🎯

Complex Fractions

A complex fraction has fractions in the numerator, denominator, or both.

Strategy: Multiply top and bottom by the LCD of all mini-fractions.

Example 5: Simplify $\dfrac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$.

Multiply numerator and denominator by $xy$:

$\frac{y + x}{y - x}$

Example 6: Simplify $\dfrac{\frac{2}{x+1}}{\frac{4}{(x+1)^2}}$.

$= \frac{2}{x+1} \cdot \frac{(x+1)^2}{4} = \frac{2(x+1)}{4} = \frac{x+1}{2}$

Rational Expression Practice 🧮

1. Simplify $\frac{x^2 - 25}{x + 5}$. The result is $x - \;?$ (enter the number)

2. $\frac{1}{3} + \frac{1}{6} = \frac{?}{6}$ (enter the numerator)

3. For $\frac{x}{x - 4}$, what value of $x$ makes the expression undefined?

Rational Expression Concepts 🔍

ACT-Style Questions 📋

📈 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 $b$.

Worked Examples

Example 1: Evaluate $\log_3 81$.

$3^4 = 81$, so $\log_3 81 = 4$.

Example 2: Simplify $\log_2 32 - \log_2 4$.

$\log_2\!\left(\frac{32}{4}\right) = \log_2 8 = 3$

Example 3: Solve $\log_5 x = 3$.

$x = 5^3 = 125$

Example 4: Solve $2^{x+1} = 16$.

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

Example 5 — Change of base: Express $\log_3 7$ using common log.

$\log_3 7 = \frac{\log 7}{\log 3} \approx \frac{0.845}{0.477} \approx 1.771$

Log Evaluation 🎯

Solving Logarithmic & Exponential Equations

Example 6: Solve $\log(x) + \log(x - 3) = 1$ (base 10).

$\log[x(x-3)] = 1 \implies x(x-3) = 10$

$x^2 - 3x - 10 = 0 \implies (x - 5)(x + 2) = 0$

$x = 5$ or $x = -2$. Since the argument of a log must be positive, $x = -2$ is extraneous. Answer: $x = 5$.

Example 7: Solve $3^x = 27^{x-2}$.

Write both sides with base 3: $3^x = (3^3)^{x-2} = 3^{3x-6}$.

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

Logarithm Practice 🧮

1. What is $\log_5 125$?

2. Solve $\log_2 x = 5$. What is $x$?

3. Simplify $\log_3 9 + \log_3 3$. (enter the numerical answer)

Log Rule Identification 🔍

ACT-Style Questions 📋

🔗 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 $a_1 = 2$, $r = 3$.

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

Partial Sums

Arithmetic series (sum of the first $n$ terms):

$S_n = \frac{n}{2}(a_1 + a_n) \quad \text{or} \quad S_n = \frac{n}{2}[2a_1 + (n-1)d]$

Geometric series (sum of the first $n$ terms):

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$

Example 3: Find the sum of the first 20 terms of $5, 8, 11, 14, \ldots$

$a_1 = 5$, $d = 3$, $a_{20} = 5 + 19(3) = 62$.

$S_{20} = \frac{20}{2}(5 + 62) = 10 \cdot 67 = 670$

Example 4: Find the sum of the first 6 terms of $4, 12, 36, 108, \ldots$

$a_1 = 4$, $r = 3$.

$S_6 = 4 \cdot \frac{1 - 3^6}{1 - 3} = 4 \cdot \frac{1 - 729}{-2} = 4 \cdot 364 = 1456$

Sequence Identification 🎯

Finding the Common Difference or Ratio

Arithmetic: $d = a_{n+1} - a_n$ (subtract consecutive terms).

Geometric: $r = \frac{a_{n+1}}{a_n}$ (divide consecutive terms).

Example 5: In the sequence $100, 90, 80, 70, \ldots$, $d = -10$.

$a_{15} = 100 + 14(-10) = 100 - 140 = -40$

Example 6: Find $d$ if $a_3 = 14$ and $a_7 = 30$ (arithmetic).

$a_7 = a_3 + 4d \implies 30 = 14 + 4d \implies d = 4$

Then $a_1 = a_3 - 2d = 14 - 8 = 6$.

Sequences Practice 🧮

1. Find the 12th term of the arithmetic sequence $4, 10, 16, 22, \ldots$

2. Find the common ratio of $5, 15, 45, 135, \ldots$

3. Find the sum of the first 10 terms of $1, 2, 3, \ldots, 10$.

Sequence & Series Concepts 🔍

ACT-Style Questions 📋

🏆 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

ACT Intermediate Algebra Tips

1. Know your formulas cold. The quadratic formula, log rules, and sequence formulas come up often, and the ACT does not give you a formula sheet.
2. Plug in answers (backsolving) when algebraic manipulation looks messy — it's often faster.
3. Watch for extraneous solutions — especially with logs (arguments must be positive) and rationals (denominators can't be zero).
4. Factor first in rational expressions — cancelling saves time.
5. Time management: ACT Math gives you 50 minutes for 45 questions (about 67 seconds each, with 4 answer choices per question). If a problem is eating well past that, mark it, guess, and move on — there is no penalty for wrong answers.
6. Discriminant shortcut: Before solving a quadratic, check $\Delta$ to see how many real answers to expect.

Mixed Practice — Set 1 🎯

Worked Mixed Problems

Problem A (Polynomials): What is the remainder when $P(x) = x^3 + 2x^2 - x - 2$ is divided by $(x + 2)$?

$P(-2) = -8 + 8 + 2 - 2 = 0$

Remainder is $0$, so $(x + 2)$ is a factor.

Problem B (Logs): Solve $\log_4(x - 1) = 2$.

$x - 1 = 4^2 = 16 \implies x = 17$

Problem C (Sequences): The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the common ratio.

$a_6 = a_3 \cdot r^3 \implies 96 = 12r^3 \implies r^3 = 8 \implies r = 2$

Mixed Practice — Fill In 🧮

1. Solve $x^2 + 2x - 15 = 0$. Enter the positive root.

2. If $\log_3 x = 4$, what is $x$?

3. Find the 7th term of the arithmetic sequence $10, 14, 18, 22, \ldots$

Topic Identification 🔍

ACT-Style Final Questions 📋