Quadratic equations worksheet for solving by completing the square, with examples and practice problems.

Let’s solve each equation step by step using completing the square.

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🔹 Question 1: Solve using completing the square

We’ll do a few examples to show the method, then list all answers.

#### Example (a): x² + 6x + 8 = 0

Step 1: Move constant to other side
→ x² + 6x = -8

Step 2: Take half of coefficient of x → 6/2 = 3 → square it → 9
Add 9 to both sides:
→ x² + 6x + 9 = -8 + 9
→ (x + 3)² = 1

Step 3: Square root both sides
→ x + 3 = ±√1 = ±1

Step 4: Solve for x
→ x = -3 + 1 = -2
→ x = -3 - 1 = -4

✔ Answer: x = -2 or x = -4

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#### Example (b): x² + 10x + 24 = 0

x² + 10x = -24
Half of 10 is 5 → 5² = 25
x² + 10x + 25 = -24 + 25 → (x+5)² = 1
x+5 = ±1 → x = -5±1 → x = -4 or -6

✔ Answer: x = -4 or x = -6

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#### Example (c): x² + 14x + 40 = 0

x² + 14x = -40
Half of 14 is 7 → 7² = 49
x² + 14x + 49 = -40 + 49 → (x+7)² = 9
x+7 = ±3 → x = -7±3 → x = -4 or -10

✔ Answer: x = -4 or x = -10

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#### Example (d): x² – 4x – 45 = 0

x² – 4x = 45
Half of -4 is -2 → (-2)² = 4
x² – 4x + 4 = 45 + 4 → (x–2)² = 49
x–2 = ±7 → x = 2±7 → x = 9 or -5

✔ Answer: x = 9 or x = -5

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#### Example (e): x² – 12x + 35 = 0

x² – 12x = -35
Half of -12 is -6 → 36
x² – 12x + 36 = -35 + 36 → (x–6)² = 1
x–6 = ±1 → x = 7 or 5

✔ Answer: x = 5 or x = 7

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#### Example (f): x² – 2x – 3 = 0

x² – 2x = 3
Half of -2 is -1 → 1
x² – 2x + 1 = 3 + 1 → (x–1)² = 4
x–1 = ±2 → x = 3 or -1

✔ Answer: x = 3 or x = -1

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#### Example (g): x² + 14x – 51 = 0

x² + 14x = 51
Half of 14 is 7 → 49
x² + 14x + 49 = 51 + 49 → (x+7)² = 100
x+7 = ±10 → x = 3 or -17

✔ Answer: x = 3 or x = -17

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#### Example (h): x² – 6x – 16 = 0

x² – 6x = 16
Half of -6 is -3 → 9
x² – 6x + 9 = 16 + 9 → (x–3)² = 25
x–3 = ±5 → x = 8 or -2

✔ Answer: x = 8 or x = -2

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#### Example (i): x² – 2x + 1 = 0

This is already a perfect square!
(x–1)² = 0 → x = 1 (double root)

✔ Answer: x = 1

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✔ Final Answers for Question 1:

(a) x = -2, -4
(b) x = -4, -6
(c) x = -4, -10
(d) x = 9, -5
(e) x = 5, 7
(f) x = 3, -1
(g) x = 3, -17
(h) x = 8, -2
(i) x = 1

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🔹 Question 2: More equations — same method

#### (a) x² + 5x + 4 = 0

x² + 5x = -4
Half of 5 is 2.5 → (2.5)² = 6.25
x² + 5x + 6.25 = -4 + 6.25 → (x + 2.5)² = 2.25
x + 2.5 = ±1.5 → x = -2.5 ± 1.5 → x = -1 or -4

✔ Answer: x = -1, -4

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#### (b) x² – 3x – 18 = 0

x² – 3x = 18
Half of -3 is -1.5 → 2.25
x² – 3x + 2.25 = 18 + 2.25 → (x – 1.5)² = 20.25
x – 1.5 = ±4.5 → x = 6 or -3

✔ Answer: x = 6, -3

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#### (c) x² + x – 12 = 0

x² + x = 12
Half of 1 is 0.5 → 0.25
x² + x + 0.25 = 12.25 → (x + 0.5)² = 12.25
x + 0.5 = ±3.5 → x = 3 or -4

✔ Answer: x = 3, -4

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#### (d) x² – 7x + 12 = 0

x² – 7x = -12
Half of -7 is -3.5 → 12.25
x² – 7x + 12.25 = -12 + 12.25 → (x – 3.5)² = 0.25
x – 3.5 = ±0.5 → x = 4 or 3

✔ Answer: x = 3, 4

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#### (e) x² – 11x + 24 = 0

x² – 11x = -24
Half of -11 is -5.5 → 30.25
x² – 11x + 30.25 = -24 + 30.25 → (x – 5.5)² = 6.25
x – 5.5 = ±2.5 → x = 8 or 3

✔ Answer: x = 3, 8

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#### (f) x² – 7x – 30 = 0

x² – 7x = 30
Half of -7 is -3.5 → 12.25
x² – 7x + 12.25 = 30 + 12.25 → (x – 3.5)² = 42.25
x – 3.5 = ±6.5 → x = 10 or -3

✔ Answer: x = 10, -3

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✔ Final Answers for Question 2:

(a) x = -1, -4
(b) x = 6, -3
(c) x = 3, -4
(d) x = 3, 4
(e) x = 3, 8
(f) x = 10, -3

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🔹 Question 3: Answers in simplified surd form

Now we leave answers as square roots if they don’t simplify nicely.

#### (a) x² + 4x – 3 = 0

x² + 4x = 3
Half of 4 is 2 → 4
x² + 4x + 4 = 3 + 4 → (x+2)² = 7
x+2 = ±√7 → x = -2 ± √7

✔ Answer: x = -2 + √7, -2 - √7

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#### (b) x² + 6x – 10 = 0

x² + 6x = 10
Half of 6 is 3 → 9
x² + 6x + 9 = 19 → (x+3)² = 19
x = -3 ± √19

✔ Answer: x = -3 + √19, -3 - √19

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#### (c) x² – 2x – 5 = 0

x² – 2x = 5
Half of -2 is -1 → 1
x² – 2x + 1 = 6 → (x–1)² = 6
x = 1 ± √6

✔ Answer: x = 1 + √6, 1 - √6

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#### (d) x² – 10x + 1 = 0

x² – 10x = -1
Half of -10 is -5 → 25
x² – 10x + 25 = 24 → (x–5)² = 24
x = 5 ± √24 → simplify √24 = 2√6
→ x = 5 ± 2√6

✔ Answer: x = 5 + 2√6, 5 - 2√6

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#### (e) x² + 8x + 3 = 0

x² + 8x = -3
Half of 8 is 4 → 16
x² + 8x + 16 = 13 → (x+4)² = 13
x = -4 ± √13

✔ Answer: x = -4 + √13, -4 - √13

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#### (f) x² – 8x – 22 = 0

x² – 8x = 22
Half of -8 is -4 → 16
x² – 8x + 16 = 38 → (x–4)² = 38
x = 4 ± √38

✔ Answer: x = 4 + √38, 4 - √38

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✔ Final Answers for Question 3:

(a) x = -2 ± √7
(b) x = -3 ± √19
(c) x = 1 ± √6
(d) x = 5 ± 2√6
(e) x = -4 ± √13
(f) x = 4 ± √38

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Final Answer:
Question 1:
(a) x = -2, -4
(b) x = -4, -6
(c) x = -4, -10
(d) x = 9, -5
(e) x = 5, 7
(f) x = 3, -1
(g) x = 3, -17
(h) x = 8, -2
(i) x = 1

Question 2:
(a) x = -1, -4
(b) x = 6, -3
(c) x = 3, -4
(d) x = 3, 4
(e) x = 3, 8
(f) x = 10, -3

Question 3:
(a) x = -2 ± √7
(b) x = -3 ± √19
(c) x = 1 ± √6
(d) x = 5 ± 2√6
(e) x = -4 ± √13
(f) x = 4 ± √38