Algebra 1 · Unit 9
Solve a quadratic by completing the square. Divide by a first when a ≠ 1.
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Make a perfect square
For x² + bx, take half of b and square it. Add that number to both sides.
Then square root
If (x + p)² = k and k > 0, x + p = ±√k.
If a ≠ 1
Divide through by a first, then complete the square.
Each line is one step
Write the original equation, then the rearranged equation, then substitute, then simplify — each on its own line.
a = 1. Integer half of b.
x² + 6x − 7 = 0
x² + 6x = 7
Half of 6 is 3, and 3² = 9.
x² + 6x + 9 = 7 + 9
(x + 3)² = 16
x + 3 = ±4
x = 1 or x = −7
Answer: x = −7, x = 1
x² + 8x = −12
Half of 8 is 4, and 4² = 16.
x² + 8x + 16 = −12 + 16
(x + 4)² = 4
x + 4 = ±2
x = −2 or x = −6
Answer: x = −6, x = −2
x² − 4x = 5
Half of −4 is −2, and (−2)² = 4.
x² − 4x + 4 = 5 + 4
(x − 2)² = 9
x − 2 = ±3
x = 5 or x = −1
Answer: x = −1, x = 5
x² + 10x = −9
Half of 10 is 5, 5² = 25.
(x + 5)² = 16
x + 5 = ±4
x = −1 or x = −9
Answer: x = −9, x = −1
Half of −2 is −1, square 1.
x² − 2x + 1 = 9
(x − 1)² = 9
x − 1 = ±3
x = 4 or x = −2
Answer: x = −2, x = 4
Half of 12 is 6, 6² = 36.
c = 36
(x + 6)²
Answer: c = 36; (x + 6)²
The squared half is a fraction, or k is not a perfect square.
x² + 8x = −9
Half of 8 is 4, 4² = 16.
x² + 8x + 16 = −9 + 16
(x + 4)² = 7
x + 4 = ±√7
x = −4 ± √7
Answer: −4 ± √7
x² + 3x = 4
Half of 3 is 32, square 94.
x² + 3x + 94 = 4 + 94
(x + 32)² = 254
x + 32 = ±52
x = 1 or x = −4
Answer: x = −4, x = 1
x² − 6x = −7
Half of −6 is −3, square 9.
(x − 3)² = 2
x = 3 ± √2
Answer: x = 3 ± √2
x² + 10x + 25 − 25 + 21
(x + 5)² − 4
Answer: (x + 5)² − 4
x² + 5x = −1
Half of 5 is 52, square 254.
(x + 52)² = 214
x = −5 ± √212
Answer: −5 ± √212
x² − x = 6
Half of −1 is −12, square 14.
(x − 12)² = 254
x − 12 = ±52
x = 3 or x = −2
Answer: x = −2, x = 3
Divide by a first, then complete the square.
2x² + 8x − 10 = 0
Divide by 2.
x² + 4x − 5 = 0
x² + 4x = 5
Half of 4 is 2, 2² = 4.
x² + 4x + 4 = 5 + 4
(x + 2)² = 9
x + 2 = ±3
x = 1 or x = −5
Answer: x = −5, x = 1
Divide by 2: x² − 6x + 5 = 0
x² − 6x = −5
(x − 3)² = 4
x − 3 = ±2
x = 5 or x = 1
Answer: x = 1, x = 5
Divide by 3: x² + 2x − 1 = 0
x² + 2x = 1
(x + 1)² = 2
x = −1 ± √2
Answer: −1 ± √2
2(x² + 6x) + 8
2(x² + 6x + 9 − 9) + 8
2((x + 3)² − 9) + 8
2(x + 3)² − 18 + 8
2(x + 3)² − 10
Answer: 2(x + 3)² − 10
x² + 23x = 1
Half of 23 is 13, square 19.
(x + 13)² = 109
x = −1 ± √103
Answer: −1 ± √103
Divide by 4: x² + x − 34 = 0
x² + x = 34
Half of 1 is 12, square 14.
(x + 12)² = 1
x + 12 = ±1
x = 12 or x = −32
Answer: x = −3/2, x = 1/2
Divide by a: x² + bax = −ca
Half of ba is b2a. Square: b²4a².
(x + b2a)² = b² − 4ac4a²
x + b2a = ±√(b² − 4ac)2a
x = −b ± √(b² − 4ac)2a
Answer: x = −b ± √(b² − 4ac)2a
Add 9 to both sides.
x² + 6x + 9 = 11
(x + 3)² = 11
x = −3 ± √11
Answer: −3 ± √11
CTS: (x + 4)² = 7, x = −4 ± √7
Formula: −8 ± √(64 − 36)2 = −8 ± √282
√28 = 2√7, so −8 ± 2√72 = −4 ± √7
Answer: −4 ± √7
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