H Algebra 2/Trig · Unit 2
Isolate a square, then take plus-or-minus square roots. Forgetting the plus or minus is the most common error.
What times itself equals 100.
Yes, positive ten times positive ten.
What else.
−10 × −10 = 100
so x² = 100
x = ±10
Forgetting plus or minus is the most common error.
An x² term and a number. Set them equal, then take plus-or-minus square roots.
x² = 81
The formula is
x = ±√k
Substitute.
x = ±√81
x = ±9
Answer: x = ±9
4x² = 36
Divide by 4.
x² = 9
The formula is
x = ±√k
Substitute.
x = ±√9
x = ±3
Answer: x = ±3
x² = −16
The formula is
x = ±√k
Substitute.
x = ±√(−16)
x = ±√16 i
x = ±4i
Answer: x = ±4i
3x² = 75
Divide by 3.
x² = 25
The formula is
x = ±√k
Substitute.
x = ±√25
x = ±5
Answer: x = ±5
2x² = −50
Divide by 2.
x² = −25
The formula is
x = ±√k
Substitute.
x = ±√(−25)
x = ±√25 i
x = ±5i
Answer: x = ±5i
A squared binomial. The last two: move the number first.
(x + 4)² = 15
The formula is
x − h = ±√k
Substitute.
x + 4 = ±√15
x = −4 ± √15
Answer: x = −4 ± √15
(x + 5)² = 7
The formula is
x − h = ±√k
Substitute.
x + 5 = ±√7
x = −5 ± √7
Answer: x = −5 ± √7
(x − 3)² = 11
The formula is
x − h = ±√k
Substitute.
x − 3 = ±√11
x = 3 ± √11
Answer: x = 3 ± √11
(x + 6)² = 20
The formula is
x − h = ±√k
Substitute.
x + 6 = ±√20
√20 = 2√5
x + 6 = ±2√5
x = −6 ± 2√5
Answer: x = −6 ± 2√5
(x − 8)² + 3 = 10
Subtract 3.
(x − 8)² = 7
The formula is
x − h = ±√k
Substitute.
x − 8 = ±√7
x = 8 ± √7
Answer: x = 8 ± √7
(x + 1)² − 6 = 2
Add 6.
(x + 1)² = 8
The formula is
x − h = ±√k
Substitute.
x + 1 = ±√8
√8 = 2√2
x + 1 = ±2√2
x = −1 ± 2√2
Answer: x = −1 ± 2√2
A coefficient in front of the squared binomial.
3(x − 6)² = 75
Divide by 3.
(x − 6)² = 25
The formula is
x − h = ±√k
Substitute.
x − 6 = ±√25
x − 6 = ±5
x = 6 + 5
x = 11
x = 6 − 5
x = 1
Answer: x = 11 and x = 1
2(x − 1)² = 18
Divide by 2.
(x − 1)² = 9
The formula is
x − h = ±√k
Substitute.
x − 1 = ±√9
x − 1 = ±3
x = 1 + 3
x = 4
x = 1 − 3
x = −2
Answer: x = 4 and x = −2
5(x + 3)² = 80
Divide by 5.
(x + 3)² = 16
The formula is
x − h = ±√k
Substitute.
x + 3 = ±√16
x + 3 = ±4
x = −3 + 4
x = 1
x = −3 − 4
x = −7
Answer: x = 1 and x = −7
3(x − 4)² = 48
Divide by 3.
(x − 4)² = 16
The formula is
x − h = ±√k
Substitute.
x − 4 = ±√16
x − 4 = ±4
x = 4 + 4
x = 8
x = 4 − 4
x = 0
Answer: x = 8 and x = 0
4(x + 2)² = 48
Divide by 4.
(x + 2)² = 12
The formula is
x − h = ±√k
Substitute.
x + 2 = ±√12
√12 = 2√3
x + 2 = ±2√3
x = −2 ± 2√3
Answer: x = −2 ± 2√3
2(x − 7)² = 14
Divide by 2.
(x − 7)² = 7
The formula is
x − h = ±√k
Substitute.
x − 7 = ±√7
x = 7 ± √7
Answer: x = 7 ± √7
Isolate the square. One or two real solutions.
3(x + 1)² − 12 = 15
Add 12.
3(x + 1)² = 27
Divide by 3.
(x + 1)² = 9
The formula is
x − h = ±√k
Substitute.
x + 1 = ±√9
x + 1 = ±3
x = −1 + 3
x = 2
x = −1 − 3
x = −4
Answer: x = 2 and x = −4
2(x − 5)² + 7 = 39
Subtract 7.
2(x − 5)² = 32
Divide by 2.
(x − 5)² = 16
The formula is
x − h = ±√k
Substitute.
x − 5 = ±√16
x − 5 = ±4
x = 5 + 4
x = 9
x = 5 − 4
x = 1
Answer: x = 9 and x = 1
4(x + 3)² − 10 = 26
Add 10.
4(x + 3)² = 36
Divide by 4.
(x + 3)² = 9
The formula is
x − h = ±√k
Substitute.
x + 3 = ±√9
x + 3 = ±3
x = −3 + 3
x = 0
x = −3 − 3
x = −6
Answer: x = 0 and x = −6
5(x − 2)² + 9 = 9
Subtract 9.
5(x − 2)² = 0
Divide by 5.
(x − 2)² = 0
The formula is
x − h = ±√k
Substitute.
x − 2 = ±√0
x − 2 = 0
x = 2
Answer: x = 2
2(x + 4)² − 5 = 11
Add 5.
2(x + 4)² = 16
Divide by 2.
(x + 4)² = 8
The formula is
x − h = ±√k
Substitute.
x + 4 = ±√8
√8 = 2√2
x + 4 = ±2√2
x = −4 ± 2√2
Answer: x = −4 ± 2√2
A negative coefficient. Imaginary solutions.
−2(x + 3)² − 5 = 11
Add 5.
−2(x + 3)² = 16
Divide by −2.
(x + 3)² = −8
The formula is
x − h = ±√k
Substitute.
x + 3 = ±√(−8)
x + 3 = ±√8 i
√8 = 2√2
x + 3 = ±2√2 i
x = −3 ± 2√2 i
Answer: x = −3 ± 2√2 i
−3(x − 4)² + 2 = 29
Subtract 2.
−3(x − 4)² = 27
Divide by −3.
(x − 4)² = −9
The formula is
x − h = ±√k
Substitute.
x − 4 = ±√(−9)
x − 4 = ±√9 i
x − 4 = ±3i
x = 4 ± 3i
Answer: x = 4 ± 3i
−4(x + 5)² − 8 = 8
Add 8.
−4(x + 5)² = 16
Divide by −4.
(x + 5)² = −4
The formula is
x − h = ±√k
Substitute.
x + 5 = ±√(−4)
x + 5 = ±√4 i
x + 5 = ±2i
x = −5 ± 2i
Answer: x = −5 ± 2i
−(x − 2)² − 3 = 6
Add 3.
−(x − 2)² = 9
Divide by −1.
(x − 2)² = −9
The formula is
x − h = ±√k
Substitute.
x − 2 = ±√(−9)
x − 2 = ±√9 i
x − 2 = ±3i
x = 2 ± 3i
Answer: x = 2 ± 3i
−5(x + 6)² + 4 = 24
Subtract 4.
−5(x + 6)² = 20
Divide by −5.
(x + 6)² = −4
The formula is
x − h = ±√k
Substitute.
x + 6 = ±√(−4)
x + 6 = ±√4 i
x + 6 = ±2i
x = −6 ± 2i
Answer: x = −6 ± 2i
One problem from each level, mixed up.
3(x + 5)² = 27
Divide by 3.
(x + 5)² = 9
The formula is
x − h = ±√k
Substitute.
x + 5 = ±√9
x + 5 = ±3
x = −5 + 3
x = −2
x = −5 − 3
x = −8
Answer: x = −2 and x = −8
9x² = −36
Divide by 9.
x² = −4
The formula is
x = ±√k
Substitute.
x = ±√(−4)
x = ±√4 i
x = ±2i
Answer: x = ±2i
4(x − 1)² − 8 = 28
Add 8.
4(x − 1)² = 36
Divide by 4.
(x − 1)² = 9
The formula is
x − h = ±√k
Substitute.
x − 1 = ±√9
x − 1 = ±3
x = 1 + 3
x = 4
x = 1 − 3
x = −2
Answer: x = 4 and x = −2
(x − 9)² = 5
The formula is
x − h = ±√k
Substitute.
x − 9 = ±√5
x = 9 ± √5
Answer: x = 9 ± √5
−2(x − 3)² − 7 = 9
Add 7.
−2(x − 3)² = 16
Divide by −2.
(x − 3)² = −8
The formula is
x − h = ±√k
Substitute.
x − 3 = ±√(−8)
x − 3 = ±√8 i
√8 = 2√2
x − 3 = ±2√2 i
x = 3 ± 2√2 i
Answer: x = 3 ± 2√2 i
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