Solving Quadratics by Factoring worksheet with eight equations to solve using factoring method.

Let’s solve each quadratic equation by factoring. We’ll go one at a time, step by step.

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Problem ①: (x + 2)(2x + 5) = 0

This is already factored! When two things multiply to zero, at least one of them must be zero.

So set each factor equal to zero:

→ x + 2 = 0 → x = -2
→ 2x + 5 = 0 → 2x = -5 → x = -5/2

✔ Solutions: x = -2 or x = -5/2

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Problem ②: (2x + 3)(4x + 3) = 0

Again, already factored. Set each part to zero:

→ 2x + 3 = 0 → 2x = -3 → x = -3/2
→ 4x + 3 = 0 → 4x = -3 → x = -3/4

✔ Solutions: x = -3/2 or x = -3/4

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Problem ③: x² = 11x - 28

First, move all terms to one side to set the equation to zero:

x² - 11x + 28 = 0

Now factor this trinomial. We need two numbers that multiply to +28 and add to -11.

Those numbers are -7 and -4, because (-7) × (-4) = 28 and (-7) + (-4) = -11.

So: (x - 7)(x - 4) = 0

Set each factor to zero:

→ x - 7 = 0 → x = 7
→ x - 4 = 0 → x = 4

✔ Solutions: x = 7 or x = 4

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Problem ④: x² + 5x + 4 = 0

Factor the trinomial. Need two numbers that multiply to +4 and add to +5.

That’s +1 and +4.

So: (x + 1)(x + 4) = 0

Set each factor to zero:

→ x + 1 = 0 → x = -1
→ x + 4 = 0 → x = -4

✔ Solutions: x = -1 or x = -4

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Problem ⑤: 10x² = 27x - 18

Move everything to left side:

10x² - 27x + 18 = 0

Now factor this. It’s a bit trickier because of the 10.

We look for two numbers that multiply to (10 × 18) = 180 and add to -27.

Try: -15 and -12 → (-15) × (-12) = 180, (-15) + (-12) = -27 ✔️

Break middle term:

10x² - 15x - 12x + 18 = 0

Group:

(10x² - 15x) + (-12x + 18) = 0
5x(2x - 3) -6(2x - 3) = 0
→ (5x - 6)(2x - 3) = 0

Set each factor to zero:

→ 5x - 6 = 0 → x = 6/5
→ 2x - 3 = 0 → x = 3/2

✔ Solutions: x = 6/5 or x = 3/2

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Problem ⑥: 8x² + 21 = -59x

Move all terms to left:

8x² + 59x + 21 = 0

Multiply 8 × 21 = 168. Find two numbers that multiply to 168 and add to 59.

Try: 56 and 3 → 56×3=168, 56+3=59 ✔️

Break middle term:

8x² + 56x + 3x + 21 = 0
Group: (8x² + 56x) + (3x + 21) = 0
8x(x + 7) + 3(x + 7) = 0
→ (8x + 3)(x + 7) = 0

Set each factor to zero:

→ 8x + 3 = 0 → x = -3/8
→ x + 7 = 0 → x = -7

✔ Solutions: x = -3/8 or x = -7

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Problem ⑦: 28x² = -96 - 184x

Move all to left:

28x² + 184x + 96 = 0

Simplify first? All divisible by 4:

Divide entire equation by 4:

7x² + 46x + 24 = 0

Now factor 7x² + 46x + 24.

Multiply 7 × 24 = 168. Find two numbers that multiply to 168 and add to 46.

Try: 42 and 4 → 42×4=168, 42+4=46 ✔️

Break middle term:

7x² + 42x + 4x + 24 = 0
Group: (7x² + 42x) + (4x + 24) = 0
7x(x + 6) + 4(x + 6) = 0
→ (7x + 4)(x + 6) = 0

Set each factor to zero:

→ 7x + 4 = 0 → x = -4/7
→ x + 6 = 0 → x = -6

✔ Solutions: x = -4/7 or x = -6

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Problem ⑧: 35x² - 22x + 7 = 4

Subtract 4 from both sides:

35x² - 22x + 3 = 0

Now factor 35x² - 22x + 3.

Multiply 35 × 3 = 105. Find two numbers that multiply to 105 and add to -22.

Try: -21 and -1 → (-21)×(-1)=105, (-21)+(-1)=-22 ✔️

Break middle term:

35x² - 21x - x + 3 = 0
Group: (35x² - 21x) + (-x + 3) = 0
7x(5x - 3) -1(5x - 3) = 0
→ (7x - 1)(5x - 3) = 0

Set each factor to zero:

→ 7x - 1 = 0 → x = 1/7
→ 5x - 3 = 0 → x = 3/5

✔ Solutions: x = 1/7 or x = 3/5

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Final Answer:
① x = -2 or x = -5/2
② x = -3/2 or x = -3/4
③ x = 7 or x = 4
④ x = -1 or x = -4
⑤ x = 6/5 or x = 3/2
⑥ x = -3/8 or x = -7
⑦ x = -4/7 or x = -6
⑧ x = 1/7 or x = 3/5