Math worksheet for completing the square with 14 quadratic expressions.
A math worksheet titled "Completing the Square" with 14 algebraic expressions to be rewritten in completed square form.
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Step-by-step solution for: Completing the Square - Minimally Different
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Show Answer Key & Explanations
Step-by-step solution for: Completing the Square - Minimally Different
You're asked to complete the square for each of these quadratic expressions. Completing the square means rewriting a quadratic expression in the form:
> a(x - h)² + k
This is especially useful for graphing parabolas or solving quadratic equations.
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## ✔ General Method:
For an expression like x² + bx, you add and subtract (b/2)² to complete the square:
> x² + bx = (x + b/2)² - (b/2)²
If there’s a constant term already, include it in the final “+k” part.
If the leading coefficient is not 1 (like 2x², 3x², etc.), you must factor it out first from the x² and x terms, then complete the square inside the parentheses.
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Let’s go one by one.
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- Take half of 6 → 3
- Square it → 9
- Add and subtract 9:
> x² + 6x = (x + 3)² - 9
✔ Answer: (x + 3)² - 9
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- Half of 8 → 4
- Square → 16
> x² + 8x = (x + 4)² - 16
✔ Answer: (x + 4)² - 16
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- Half of -8 → -4
- Square → 16
> x² - 8x = (x - 4)² - 16
✔ Answer: (x - 4)² - 16
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- Half of -9 → -9/2
- Square → 81/4
> x² - 9x = (x - 9/2)² - 81/4
✔ Answer: (x - 9/2)² - 81/4
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- Half of -3 → -3/2
- Square → 9/4
> x² - 3x = (x - 3/2)² - 9/4
✔ Answer: (x - 3/2)² - 9/4
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We already know x² - 3x = (x - 3/2)² - 9/4
So:
> x² - 3x - 1 = (x - 3/2)² - 9/4 - 1 = (x - 3/2)² - 13/4
✔ Answer: (x - 3/2)² - 13/4
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Factor out 2 from the first two terms:
> 2(x² + 3x) + 2
Now complete the square inside:
- Half of 3 → 3/2
- Square → 9/4
So:
> 2[(x + 3/2)² - 9/4] + 2
> = 2(x + 3/2)² - 18/4 + 2
> = 2(x + 3/2)² - 9/2 + 4/2
> = 2(x + 3/2)² - 5/2
✔ Answer: 2(x + 3/2)² - 5/2
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Factor out 2:
> 2(x² + (5/2)x) + 2
Half of 5/2 → 5/4
Square → 25/16
> 2[(x + 5/4)² - 25/16] + 2
> = 2(x + 5/4)² - 50/16 + 2
> = 2(x + 5/4)² - 25/8 + 16/8
> = 2(x + 5/4)² - 9/8
✔ Answer: 2(x + 5/4)² - 9/8
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Factor out 2:
> 2(x² + (1/2)x) + 2
Half of 1/2 → 1/4
Square → 1/16
> 2[(x + 1/4)² - 1/16] + 2
> = 2(x + 1/4)² - 2/16 + 2
> = 2(x + 1/4)² - 1/8 + 16/8
> = 2(x + 1/4)² + 15/8
✔ Answer: 2(x + 1/4)² + 15/8
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Factor out 3:
> 3(x² + (1/3)x) + 2
Half of 1/3 → 1/6
Square → 1/36
> 3[(x + 1/6)² - 1/36] + 2
> = 3(x + 1/6)² - 3/36 + 2
> = 3(x + 1/6)² - 1/12 + 24/12
> = 3(x + 1/6)² + 23/12
✔ Answer: 3(x + 1/6)² + 23/12
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Same as above, but constant is -10:
> 3(x² + (1/3)x) - 10
> = 3[(x + 1/6)² - 1/36] - 10
> = 3(x + 1/6)² - 3/36 - 10
> = 3(x + 1/6)² - 1/12 - 120/12
> = 3(x + 1/6)² - 121/12
✔ Answer: 3(x + 1/6)² - 121/12
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Factor out 4:
> 4(x² + (1/4)x) - 10
Half of 1/4 → 1/8
Square → 1/64
> 4[(x + 1/8)² - 1/64] - 10
> = 4(x + 1/8)² - 4/64 - 10
> = 4(x + 1/8)² - 1/16 - 160/16
> = 4(x + 1/8)² - 161/16
✔ Answer: 4(x + 1/8)² - 161/16
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## 📌 Final Answers Summary:
1. (x + 3)² - 9
2. (x + 4)² - 16
3. (x - 4)² - 16
4. (x - 9/2)² - 81/4
5. (x - 3/2)² - 9/4
6. (x - 3/2)² - 13/4
7. 2(x + 3/2)² - 5/2
8. 2(x + 5/4)² - 9/8
9. 2(x + 1/4)² + 15/8
10. 3(x + 1/6)² + 23/12
11. 3(x + 1/6)² - 121/12
12. 4(x + 1/8)² - 161/16
---
Let me know if you’d like to see any step explained in more detail!
> a(x - h)² + k
This is especially useful for graphing parabolas or solving quadratic equations.
---
## ✔ General Method:
For an expression like x² + bx, you add and subtract (b/2)² to complete the square:
> x² + bx = (x + b/2)² - (b/2)²
If there’s a constant term already, include it in the final “+k” part.
If the leading coefficient is not 1 (like 2x², 3x², etc.), you must factor it out first from the x² and x terms, then complete the square inside the parentheses.
---
Let’s go one by one.
---
1. x² + 6x
- Take half of 6 → 3
- Square it → 9
- Add and subtract 9:
> x² + 6x = (x + 3)² - 9
✔ Answer: (x + 3)² - 9
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2. x² + 8x
- Half of 8 → 4
- Square → 16
> x² + 8x = (x + 4)² - 16
✔ Answer: (x + 4)² - 16
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3. x² - 8x
- Half of -8 → -4
- Square → 16
> x² - 8x = (x - 4)² - 16
✔ Answer: (x - 4)² - 16
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4. x² - 9x
- Half of -9 → -9/2
- Square → 81/4
> x² - 9x = (x - 9/2)² - 81/4
✔ Answer: (x - 9/2)² - 81/4
---
5. x² - 3x
- Half of -3 → -3/2
- Square → 9/4
> x² - 3x = (x - 3/2)² - 9/4
✔ Answer: (x - 3/2)² - 9/4
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6. x² - 3x - 1
We already know x² - 3x = (x - 3/2)² - 9/4
So:
> x² - 3x - 1 = (x - 3/2)² - 9/4 - 1 = (x - 3/2)² - 13/4
✔ Answer: (x - 3/2)² - 13/4
---
7. 2x² + 6x + 2
Factor out 2 from the first two terms:
> 2(x² + 3x) + 2
Now complete the square inside:
- Half of 3 → 3/2
- Square → 9/4
So:
> 2[(x + 3/2)² - 9/4] + 2
> = 2(x + 3/2)² - 18/4 + 2
> = 2(x + 3/2)² - 9/2 + 4/2
> = 2(x + 3/2)² - 5/2
✔ Answer: 2(x + 3/2)² - 5/2
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8. 2x² + 5x + 2
Factor out 2:
> 2(x² + (5/2)x) + 2
Half of 5/2 → 5/4
Square → 25/16
> 2[(x + 5/4)² - 25/16] + 2
> = 2(x + 5/4)² - 50/16 + 2
> = 2(x + 5/4)² - 25/8 + 16/8
> = 2(x + 5/4)² - 9/8
✔ Answer: 2(x + 5/4)² - 9/8
---
9. 2x² + x + 2
Factor out 2:
> 2(x² + (1/2)x) + 2
Half of 1/2 → 1/4
Square → 1/16
> 2[(x + 1/4)² - 1/16] + 2
> = 2(x + 1/4)² - 2/16 + 2
> = 2(x + 1/4)² - 1/8 + 16/8
> = 2(x + 1/4)² + 15/8
✔ Answer: 2(x + 1/4)² + 15/8
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10. 3x² + x + 2
Factor out 3:
> 3(x² + (1/3)x) + 2
Half of 1/3 → 1/6
Square → 1/36
> 3[(x + 1/6)² - 1/36] + 2
> = 3(x + 1/6)² - 3/36 + 2
> = 3(x + 1/6)² - 1/12 + 24/12
> = 3(x + 1/6)² + 23/12
✔ Answer: 3(x + 1/6)² + 23/12
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11. 3x² + x - 10
Same as above, but constant is -10:
> 3(x² + (1/3)x) - 10
> = 3[(x + 1/6)² - 1/36] - 10
> = 3(x + 1/6)² - 3/36 - 10
> = 3(x + 1/6)² - 1/12 - 120/12
> = 3(x + 1/6)² - 121/12
✔ Answer: 3(x + 1/6)² - 121/12
---
12. 4x² + x - 10
Factor out 4:
> 4(x² + (1/4)x) - 10
Half of 1/4 → 1/8
Square → 1/64
> 4[(x + 1/8)² - 1/64] - 10
> = 4(x + 1/8)² - 4/64 - 10
> = 4(x + 1/8)² - 1/16 - 160/16
> = 4(x + 1/8)² - 161/16
✔ Answer: 4(x + 1/8)² - 161/16
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## 📌 Final Answers Summary:
1. (x + 3)² - 9
2. (x + 4)² - 16
3. (x - 4)² - 16
4. (x - 9/2)² - 81/4
5. (x - 3/2)² - 9/4
6. (x - 3/2)² - 13/4
7. 2(x + 3/2)² - 5/2
8. 2(x + 5/4)² - 9/8
9. 2(x + 1/4)² + 15/8
10. 3(x + 1/6)² + 23/12
11. 3(x + 1/6)² - 121/12
12. 4(x + 1/8)² - 161/16
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Let me know if you’d like to see any step explained in more detail!
Parent Tip: Review the logic above to help your child master the concept of completing the square worksheet.
