Quadratic Equations and Factoring — ACT Math Guide

Q: What You Need to Know

• Standard form: ax² + bx + c = 0 (where a ≠ 0) • Factored form: a(x - r)(x - s) = 0 where r and s are roots • Three main solving methods: factoring, quadratic formula, completing the square • Zero product property: if ab = 0, then a = 0 or b = 0 • Discriminant b² - 4ac determines number and type of

Quadratic equations ACT problems appear frequently throughout the math section, making up about 4-6 questions out of 60 total. These equations involve variables raised to the second power and can be solved through factoring, completing the square, or using the quadratic formula. With 60 questions in 60 minutes on the ACT, knowing multiple solution methods helps you pick the fastest approach. You've got this — quadratics become much easier once you recognize the patterns!

What You Need to Know

Standard form: ax² + bx + c = 0 (where a ≠ 0)

Factored form: a(x - r)(x - s) = 0 where r and s are roots

Three main solving methods: factoring, quadratic formula, completing the square

Zero product property: if ab = 0, then a = 0 or b = 0

Discriminant b² - 4ac determines number and type of solutions

Parabola vertex form: y = a(x - h)² + k where (h, k) is the vertex

📐 KEY FORMULA: x = (-b ± √(b² - 4ac)) / (2a)

⏱️ ACT TIME TIP: Try factoring first — it's usually faster than the quadratic formula when it works cleanly

How to Solve Quadratic Equations on the ACT

Example Question 1 — Easy/Medium Difficulty

Which of the following is a solution to x² - 5x - 6 = 0?

A) -6

B) -1

C) 1

D) 5

E) 6

Solution:

Step 1: Factor the quadratic by finding two numbers that multiply to -6 and add to -5

Step 2: Those numbers are -6 and +1, so: (x - 6)(x + 1) = 0

Step 3: Use zero product property: x - 6 = 0 or x + 1 = 0, giving x = 6 or x = -1

✅Answer: E — Both x = 6 and x = -1 are solutions, and only 6 appears in the choices

Example Question 2 — Hard Difficulty

If 2x² - 8x + 3 = 0, what is the sum of the two solutions?

A) -4

B) -2

C) 2

D) 4

E) 8

Solution:

Step 1: Use Vieta's formulas — for ax² + bx + c = 0, sum of roots = -b/a

Step 2: Here a = 2 and b = -8, so sum = -(-8)/2 = 8/2 = 4

Step 3: Verify with quadratic formula if needed, but Vieta's is much faster

✅Answer: D — The sum of solutions equals -b/a = 4

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting to check both solutions in the original equation

✅Fix: Always substitute back, especially when you squared both sides

❌Mistake: Using the quadratic formula when simple factoring works

✅Fix: Look for easy factors first — save time for harder questions later

❌Mistake: Mixing up the signs in factoring (x + 3)(x - 2) vs (x - 3)(x + 2)

✅Fix: FOIL your factored form to double-check: multiply it out quickly

❌Mistake: Stopping after finding one solution when the question asks for both

✅Fix: Read carefully — ACT math questions often want specific information about solutions

Practice Question — Try It Yourself

What are the solutions to 3x² + 7x - 6 = 0?

A) x = -3 and x = 2/3

B) x = -3 and x = -2/3

C) x = 3 and x = -2/3

D) x = -3 and x = 2/3

E) x = 3 and x = 2/3

Show Answer

Answer: A — Factor as (3x - 2)(x + 3) = 0, giving x = 2/3 and x = -3

Key Takeaways for the ACT

Try factoring first — it's faster than the quadratic formula for most ACT problems

Remember Vieta's formulas for finding sum and product of roots without solving

The discriminant b² - 4ac tells you about solutions: positive = 2 real, zero = 1 real, negative = 0 real

Don't forget the zero product property works both ways

ACT math quadratic equations often connect to graphing parabolas and finding intercepts