How to Solve: ACT Math – Quadratic Equations and Functions

How to Solve: ACT Math – Quadratic Equations and Functions

Introduction

Mastering quadratics on the ACT can boost your math score by 3–5 points—because they appear in 5–7 questions per test. Whether you’re factoring, graphing, or solving word problems, this guide gives you the exact steps to solve any quadratic question in under 60 seconds.

What You Need To Know First

Before diving in, make sure you understand: 1. Factoring (e.g., turning x² + 5x + 6 into (x + 2)(x + 3)). 2. Square roots (e.g., √9 = 3, and √(x²) = |x|). 3. Basic graphing (how to plot points and recognize parabolas).

If any of these feel shaky, pause and review them first.

Key Vocabulary

Formulas To Know

Note: The ACT provides the quadratic formula, but you must know how to use it quickly.

Step-by-Step Method

How to Solve Any Quadratic Equation (5 Steps)

1. Write the equation in standard form (ax² + bx + c = 0).
2. Check if it can be factored easily (look for two numbers that multiply to ac and add to b).
3. If factoring is hard, use the quadratic formula (x = [–b ± √(b² – 4ac)] / (2a)).
4. Simplify the roots (split into two answers if there’s a ±).
5. Check your answers (plug them back into the original equation).

How to Graph a Quadratic Function (4 Steps)

1. Find the vertex (use x = –b/(2a) for standard form, or (h, k) for vertex form).
2. Find the axis of symmetry (the vertical line x = h).
3. Find the roots (set y = 0 and solve for x).
4. Plot the vertex, roots, and 1–2 extra points (pick x = 1 and x = –1 if needed).

Worked Examples

Example 1 – Basic (Factoring)

Problem: Solve x² – 5x + 6 = 0.

Step 1: Already in standard form (ax² + bx + c = 0). Step 2: Factor. Find two numbers that multiply to 6 and add to –5 → –2 and –3. Step 3: Write factored form: (x – 2)(x – 3) = 0. Step 4: Set each factor to zero: x – 2 = 0 → x = 2; x – 3 = 0 → x = 3. Answer: x = 2 and x = 3.

What we did and why: Factoring is the fastest method when possible. We split the quadratic into two binomials and solved for x where each equals zero.

Example 2 – Medium (Quadratic Formula)

Problem: Solve 2x² + 4x – 3 = 0.

Step 1: Already in standard form (a = 2, b = 4, c = –3). Step 2: Factoring is messy (try it—no easy pairs). Use the quadratic formula. Step 3: Plug into x = [–b ± √(b² – 4ac)] / (2a):
- b² – 4ac = 16 – 4(2)(–3) = 16 + 24 = 40.
- √40 = 2√10.
- x = [–4 ± 2√10] / 4. Step 4: Simplify: x = –1 ± (√10)/2. Answer: x = –1 + (√10)/2 and x = –1 – (√10)/2.

What we did and why: When factoring fails, the quadratic formula always works. We simplified the square root and reduced the fraction.

Example 3 – Exam-Style (Word Problem)

Problem: A ball is thrown upward from the ground with an initial velocity of 48 ft/s. Its height h (in feet) after t seconds is h = –16t² + 48t. When does the ball hit the ground?

Step 1: The ball hits the ground when h = 0. So, –16t² + 48t = 0. Step 2: Factor out –16t: –16t(t – 3) = 0. Step 3: Set each factor to zero: –16t = 0 → t = 0; t – 3 = 0 → t = 3. Step 4: t = 0 is when the ball is thrown. t = 3 is when it lands. Answer: The ball hits the ground at t = 3 seconds.

What we did and why: We translated the word problem into an equation, factored, and discarded the irrelevant solution (t = 0).

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Hey—quadratics on the ACT? You’ve got this. Here’s the game plan: 1. If it’s an equation, write it as ax² + bx + c = 0. 2. Try factoring first—if it’s easy, do it. If not, use the quadratic formula. 3. For graphs, find the vertex with x = –b/(2a), then plot roots and symmetry. 4. Watch for traps: Hidden x², negative signs, and word problems with two answers (only one makes sense). 5. Double-check your work—plug answers back in or simplify radicals fully.

You’ll see 5–7 of these on test day. Nail them, and you’re 3–5 points closer to your goal. Now go crush it!