Let’s solve each quadratic equation step by step.
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1. 2x² – 2x – 24 = 0
First, simplify by dividing all terms by 2:
→ x² – x – 12 = 0
Now factor: We need two numbers that multiply to -12 and add to -1.
Those are -4 and +3.
→ (x – 4)(x + 3) = 0
So, x = 4 or x = -3
✔ Check:
(4)² – 4 – 12 = 16 – 4 – 12 = 0 ✔️
(-3)² – (-3) – 12 = 9 + 3 – 12 = 0 ✔️
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2. x² + 10x + 16 = 0
Factor: Need two numbers that multiply to 16 and add to 10 → 8 and 2.
→ (x + 8)(x + 2) = 0
So, x = -8 or x = -2
✔ Check:
(-8)² + 10*(-8) + 16 = 64 – 80 + 16 = 0 ✔️
(-2)² + 10*(-2) + 16 = 4 – 20 + 16 = 0 ✔️
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3. 4x² + 15x + 9 = 0
Use factoring by grouping or quadratic formula. Let’s try factoring.
We need two numbers that multiply to 4*9=36 and add to 15 → 12 and 3.
Rewrite middle term:
→ 4x² + 12x + 3x + 9 = 0
Group:
→ (4x² + 12x) + (3x + 9) = 0
→ 4x(x + 3) + 3(x + 3) = 0
→ (4x + 3)(x + 3) = 0
So, x = -3/4 or x = -3
✔ Check:
x = -3: 4*(9) + 15*(-3) + 9 = 36 – 45 + 9 = 0 ✔️
x = -3/4: 4*(9/16) + 15*(-3/4) + 9 = 9/4 – 45/4 + 36/4 = (9 - 45 + 36)/4 = 0/4 = 0 ✔️
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4. x² – 13x + 42 = 0
Factor: Two numbers that multiply to 42 and add to -13 → -6 and -7
→ (x – 6)(x – 7) = 0
So, x = 6 or x = 7
✔ Check:
6² – 13*6 + 42 = 36 – 78 + 42 = 0 ✔️
7² – 13*7 + 42 = 49 – 91 + 42 = 0 ✔️
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5. 6x² – x – 40 = 0
Try factoring. Multiply 6 * (-40) = -240. Find two numbers that multiply to -240 and add to -1 → -16 and +15.
Rewrite:
→ 6x² – 16x + 15x – 40 = 0
Group:
→ (6x² – 16x) + (15x – 40) = 0
→ 2x(3x – 8) + 5(3x – 8) = 0
→ (2x + 5)(3x – 8) = 0
So, x = -5/2 or x = 8/3
✔ Check:
x = -5/2: 6*(25/4) – (-5/2) – 40 = 150/4 + 5/2 – 40 = 75/2 + 5/2 – 80/2 = (80 - 80)/2 = 0 ✔️
x = 8/3: 6*(64/9) – (8/3) – 40 = 384/9 – 24/9 – 360/9 = (384 - 24 - 360)/9 = 0/9 = 0 ✔️
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6. 2x² + 13x + 15 = 0
Multiply 2 * 15 = 30. Find two numbers that multiply to 30 and add to 13 → 10 and 3.
Rewrite:
→ 2x² + 10x + 3x + 15 = 0
Group:
→ (2x² + 10x) + (3x + 15) = 0
→ 2x(x + 5) + 3(x + 5) = 0
→ (2x + 3)(x + 5) = 0
So, x = -3/2 or x = -5
✔ Check:
x = -3/2: 2*(9/4) + 13*(-3/2) + 15 = 9/2 – 39/2 + 30/2 = (9 - 39 + 30)/2 = 0 ✔️
x = -5: 2*25 + 13*(-5) + 15 = 50 – 65 + 15 = 0 ✔️
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Final Answer:
x = 4, -3; x = -8, -2; x = -3/4, -3; x = 6, 7; x = -5/2, 8/3; x = -3/2, -5
Parent Tip: Review the logic above to help your child master the concept of quadratic formula practice problems worksheet.
