Solving Quadratic Equations by Factoring- Practice Worksheet

Solving Quadratic Equations by Factoring — Practice Worksheet

Factoring quadratics is the fastest way to solve equations when it works. It doesn't work on everything, but when it does, you get exact answers in seconds. No calculators. No decimals. Just clean, integer solutions.

This post is a straight shot: how factoring works, how to do it step by step, and a pile of practice problems you can actually use. Print it, copy it, work it. Let's go.

What Factoring Actually Means

A quadratic equation is any equation that looks like ax² + bx + c = 0. The "a" can't be zero, or it's not quadratic anymore.

Factoring means breaking that polynomial into two binomials multiplied together. If you can rewrite ax² + bx + c as (something)(something else) = 0, you win.

Here's why: the Zero Product Property. If two things multiply to zero, at least one of them is zero. So you set each factor equal to zero and solve. Done.

How to Factor — Step by Step

Step 1: Get Everything on One Side

Move every term to the left side so the right side is zero. If your equation is x² = 5x - 6, rewrite it as x² - 5x + 6 = 0.

Step 2: Look for a Greatest Common Factor (GCF)

Always check first. Can you pull out a number or variable from every term?

2x² + 8x = 0 becomes 2x(x + 4) = 0. Set each factor to zero: x = 0 or x = -4. That's it.

Step 3: Factor the Trinomial

No GCF? Time to split the middle term. For x² - 5x + 6, you need two numbers that:

• Multiply to 6 (the constant term)
• Add to -5 (the middle coefficient)

Those numbers are -2 and -3. So you write (x - 2)(x - 3) = 0.

Step 4: Set Each Factor to Zero

x - 2 = 0 gives x = 2. x - 3 = 0 gives x = 3.

Step 5: Check Your Answers

Plug them back in. (2)² - 5(2) + 6 = 4 - 10 + 6 = 0. (3)² - 5(3) + 6 = 9 - 15 + 6 = 0. Both work.

Factoring Methods Compared

Not every quadratic factors the same way. Here's a quick look at your options:

Practice Worksheet — Set A (Beginner)

These all have a = 1. Find two numbers that multiply to c and add to b.

1. x² + 5x + 6 = 0
2. x² - 7x + 12 = 0
3. x² + 3x - 10 = 0
4. x² - x - 20 = 0
5. x² + 8x + 16 = 0

Answer Key — Set A

• 1. (x + 2)(x + 3) = 0 → x = -2, -3
• 2. (x - 3)(x - 4) = 0 → x = 3, 4
• 3. (x + 5)(x - 2) = 0 → x = -5, 2
• 4. (x - 5)(x + 4) = 0 → x = 5, -4
• 5. (x + 4)² = 0 → x = -4 (double root)

Practice Worksheet — Set B (Intermediate)

These have a ≠ 1. Use the AC method: multiply a and c, find factors that add to b, then split and factor by grouping.

1. 2x² + 7x + 3 = 0
2. 3x² - 10x + 8 = 0
3. 5x² + 13x - 6 = 0
4. 4x² - 12x + 9 = 0
5. 6x² + x - 12 = 0

Answer Key — Set B

• 1. (2x + 1)(x + 3) = 0 → x = -½, -3
• 2. (3x - 4)(x - 2) = 0 → x = 4/3, 2
• 3. (5x - 2)(x + 3) = 0 → x = 2/5, -3
• 4. (2x - 3)² = 0 → x = 3/2 (double root)
• 5. (2x + 3)(3x - 4) = 0 → x = -3/2, 4/3

Practice Worksheet — Set C (Mixed + GCF)

Some of these need a GCF pulled out first. Don't skip that step.

1. 3x² - 12x = 0
2. x² - 25 = 0
3. 2x² - 18 = 0
4. 4x² + 20x + 24 = 0
5. x² + 10x + 25 = 0

Answer Key — Set C

• 1. 3x(x - 4) = 0 → x = 0, 4
• 2. (x + 5)(x - 5) = 0 → x = -5, 5
• 3. 2(x + 3)(x - 3) = 0 → x = -3, 3
• 4. 4(x + 2)(x + 3) = 0 → x = -2, -3 (GCF was 4)
• 5. (x + 5)² = 0 → x = -5 (double root)

Common Mistakes That Waste Points

Students mess this up in the same ways every time. Avoid these:

• Forgetting to set the equation to zero first. You can't factor x² + 5x = 6 directly. Move the 6.
• Missing the GCF. If every term is even, pull out the 2. It makes the rest easier.
• Sign errors. If c is positive, both numbers have the same sign as b. If c is negative, they have opposite signs.
• Stopping at the factors. The problem asks for x, not the factored form. Solve each factor.
• Ignoring double roots. (x - 3)² = 0 has one solution: x = 3. Don't list it twice like it's two different answers.

When Factoring Doesn't Work

Not every quadratic factors nicely. If the AC method gives you ugly fractions or nothing seems to multiply and add correctly, the polynomial might be prime. That means it doesn't factor over the integers.

Your backup plans:

• Quadratic Formula: Works on everything. Ugly but reliable.
• Completing the Square: Good when you need vertex form.
• Graphing: Fast if you have a calculator and only need decimal approximations.

Factoring is a shortcut. When it fails, switch tools. No shame in that.

Real Talk: Why This Matters

You won't factor quadratics for fun as an adult. But you'll use the logic everywhere: optimization, physics, engineering, finance. The skill is recognizing structure and breaking problems into smaller pieces.

That's it. Work the problems. Check your answers. Fix your mistakes. Move on. 🔥