Let’s solve each quadratic equation by factoring. We’ll move all terms to one side so the equation equals zero, then factor and use the Zero Product Property (if ab = 0, then a=0 or b=0).
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Problem ①: x² + 5x - 35 = 3x
Step 1: Subtract 3x from both sides
→ x² + 2x - 35 = 0
Step 2: Factor the quadratic
We need two numbers that multiply to -35 and add to 2 → 7 and -5
→ (x + 7)(x - 5) = 0
Step 3: Set each factor equal to zero
x + 7 = 0 → x = -7
x - 5 = 0 → x = 5
✔ Final Answer for ①:
x = -7, 5
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Problem ②: 2x² - 14 = -3x
Step 1: Add 3x to both sides
→ 2x² + 3x - 14 = 0
Step 2: Factor using AC method
A=2, C=-14 → AC = -28
Find two numbers that multiply to -28 and add to 3 → 7 and -4
Rewrite middle term:
2x² + 7x - 4x - 14 = 0
Group: (2x² + 7x) + (-4x -14) = 0
Factor groups: x(2x+7) -2(2x+7) = 0
→ (x - 2)(2x + 7) = 0
Step 3: Solve
x - 2 = 0 → x = 2
2x + 7 = 0 → x = -7/2
✔ Final Answer for ②:
x = 2, -7/2
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Problem ③: x² = 11x - 28
Step 1: Move all terms to left
→ x² - 11x + 28 = 0
Step 2: Factor
Need two numbers that multiply to 28 and add to -11 → -4 and -7
→ (x - 4)(x - 7) = 0
Step 3: Solve
x - 4 = 0 → x = 4
x - 7 = 0 → x = 7
✔ Final Answer for ③:
x = 4, 7
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Problem ④: x² + 17x + 49 = 3x
Step 1: Subtract 3x
→ x² + 14x + 49 = 0
Step 2: Recognize perfect square trinomial
(x + 7)² = 0
Step 3: Solve
x + 7 = 0 → x = -7 (double root)
✔ Final Answer for ④:
x = -7
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Problem ⑤: (4x + 5)(x + 1) = 0
Already factored! Just set each factor to zero.
4x + 5 = 0 → x = -5/4
x + 1 = 0 → x = -1
✔ Final Answer for ⑤:
x = -5/4, -1
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Problem ⑥: x² = 2x
Step 1: Move all terms to left
→ x² - 2x = 0
Step 2: Factor out GCF
x(x - 2) = 0
Step 3: Solve
x = 0
x - 2 = 0 → x = 2
✔ Final Answer for ⑥:
x = 0, 2
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Problem ⑦: 4x² = 8x
Step 1: Move all terms to left
→ 4x² - 8x = 0
Step 2: Factor out GCF
4x(x - 2) = 0
Step 3: Solve
4x = 0 → x = 0
x - 2 = 0 → x = 2
✔ Final Answer for ⑦:
x = 0, 2
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Problem ⑧: x² + 8x = -15
Step 1: Add 15 to both sides
→ x² + 8x + 15 = 0
Step 2: Factor
Need two numbers that multiply to 15 and add to 8 → 3 and 5
→ (x + 3)(x + 5) = 0
Step 3: Solve
x + 3 = 0 → x = -3
x + 5 = 0 → x = -5
✔ Final Answer for ⑧:
x = -3, -5
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Final Answer:
① x = -7, 5
② x = 2, -7/2
③ x = 4, 7
④ x = -7
⑤ x = -5/4, -1
⑥ x = 0, 2
⑦ x = 0, 2
⑧ x = -3, -5
Parent Tip: Review the logic above to help your child master the concept of factoring quadratic equation worksheet.
