Section 5.1
1) 𝑥 2 =
16
3) 𝑏 2 = −49
√𝑏 2 = ±√−49
𝑏 = ±7𝑖
take square roots: √𝑥 2 = ±√16
simplify: 𝑥 = ±4
Solution: 𝒃 = ±𝟕𝒊
Solution: 𝒙 = ±𝟒
Check x = 4
Check x = -4
(4)2 = 16
16 = 16
(-4)2 = 16
16 = 16
Check b = 7i
Check b = -7i
(7i)2 = -49
49i2 = -49
49(-1) = 49
-49 = -49
(-7i)2 = -49
49i2 = -49
49(-1) = -49
-49 = -49
x = 4 checks and is a x = -4 checks and is a b = 7i checks and is a b = -7i checks and is a solution.
solution.
solution.
solution.
5) 𝑚2 = 98
9) (𝑥 − 3)2 =
25
√𝑚2 = ±√98
𝑚 = ±7√2
√(𝑥 − 3)2 = ±√25
𝑥 − 3 = ±5
Solution: 𝒎 = ±𝟕√𝟐
x – 3 = 5,
Check 𝑚 = 7√2
2
Check 𝑚 = −7√2
2
(7√2) = 98
(−7√2) = 98
49√4 = 98
49√4 = 98
49(2) = 98
49(2) = 98
98 = 98
98 = 98
m = 7√2 checks and m = −7√2 checks and
is a solution.
is a solution
or
x – 3 = -5
x = 8, or x = -2
Solution: x = 8, -2
Check x = 8
Check x = -2
(8-3)2 = 25
52 = 25
25=25
(-2 – 3)2 =25
(-5)2 = 25
25 = 25
x = 8 checks and x = -2 checks and
is a solution.
is a solution.
15) (2𝑥 − 6)2 = −75
2
11) (2𝑥 − 5) = 49
√(2𝑥 − 6)2 = ±√−75
√(2𝑥 − 5)2 = ±√49
2𝑥 − 6 = ±5𝑖√3
2𝑥 − 5 = ±7
2x – 5 = 7
or
2x – 5 = -7
2x = 12
or 2x = -2
2𝑥 = 6 ± 5𝑖√3
𝑥=
6±5𝑖√3
2
x = 6 or x = -1
These solutions will check,
but I am not showing the work.
These solutions will check,
but I am not showing the work.
It is harder to check these answers
than it is to find these answers.
Solution: x = 6, -1
Solution: 𝒙 =
𝟔±𝟓𝒊√𝟑
𝟐
2 2
19) (𝑥 + 3) = 21
21) x2 + 6x + C
2
√(𝑥 + 2) = ±√21
3
2
𝑥 + 3 = ±√21
2
𝑥 = − 3 ± √21 (this is an acceptable answer)
2
𝑥 = −3 +
𝑥=
3
3
Rewrite original problem with C = 9,
then factor.
x2 + 6x + 9 = (x+3)(x+3) or (x+3)2
Solution: C = 9, (x+3)2
(this is another acceptable answer)
These solutions will check,
but I am not showing the work.
It is harder to check these answers than
it is to find these answers.
𝟐
Solution: 𝒙 = − 𝟑 ± √𝟐𝟏
−𝟐±𝟑√𝟐𝟏
𝟑
25) b2 – 4b
+C
1
2
3√21
−2±3√21
or 𝒙 =
1
𝐶 = (2 ∗ 6) = 32 = 9
2
𝐶 = (2 ∗ −4) = (−2)2 = 4
Rewrite original problem with C = 4 then factor.
b2 – 4b + 4 = (b–2)(b–2) = (b–2)2
Solution: C = 4, (b – 2)2
1
33) x2 – 7x + C
37) 𝑏 2 + 2 𝑏 + 𝐶
2
1
−7 2
𝐶 = (2 ∗ −7) = ( 2 ) =
49
1
Rewrite original problem with
49
C = 4 then factor.
𝑥 2 − 7𝑥 +
49
4
7
𝟒𝟗
𝟒
7
7 2
𝑏2 +
1
2
𝐶 = (2 ∗ 6) = 32 = 9
1
16
1
1
1
1 2
𝑏 + 16 = (𝑏 + 4) (𝑏 + 4) = (𝑏 + 4)
2
𝟕 𝟐
, (𝒙 − 𝟐)
39) x2 + 6x =
7
1
1 2
Rewrite original problem with
1
C = 16 then factor.
= (𝑥 − 2) (𝑥 − 2) = (𝑥 − 2)
Solution: 𝑪 =
1 2
𝐶 = (2 ∗ 2) = (4) =
4
Solution: 𝑪 =
𝟏
𝟏 𝟐
, (𝒃 + 𝟒)
𝟏𝟔
41) a2 + 10a – 24=0
𝑎2 + 10𝑎 = 24
(rewrite with the constant isolated)
2
1
x2 + 6x + 9 = 7 + 9
𝐶 = (2 ∗ 10) = 52 = 25
(x+3)2 = 16
𝑎2 + 10𝑎 + 25 = 24 + 25
√(𝑥 + 3)2 = ±√16
(𝑎 + 5)2 = 49
𝑥 + 3 = ±4
√(𝑎 + 5)2 = ±√49
x+3 = 4
or x + 3 = -4
𝑎 + 5 = ±7
x=1
or x = -7
a+5 = 7
or a+5 = -7
These solutions will check.
I am not showing the work.
a = 2 or a = -12
Solution: x = -7, 1
These solutions will check.
I am not showing the work.
Solution: a = -12, 2
45) x2 – 8x+ 7 = 0
𝑥 2 − 8𝑥 = −7 (I must first isolate the constant)
2
1
𝐶 = (2 ∗ −8) = (−4)2 = 16
𝑥 2 − 8𝑥 + 16 = −7 + 16
(𝑥 − 4)2 = 9
√(𝑥 − 4)2 = ±√9
𝑥 − 4 = ±3
x-4 = 3
or x – 4 = -3
x = 7 or x = 1
These solutions will check.
I am not showing the work.
Solution: x = 7, 1
51) x2 + 6x =
5
55) x2 + 8x = -20
1
2
𝐶 = (2 ∗ 8) = 42 = 16
2
1
𝑥 2 + 8𝑥 + 16 = −20 + 16
2
𝐶 = (2 ∗ 6) = 3 = 9
(𝑥 + 4)2 = −4
2
x + 6x + 9 = 5 + 9
√(𝑥 + 4)2 = ±√−4
(x+3)2 = 14
𝑥 + 4 = ±2𝑖
𝑥 + 3 = ±√14
𝑥 = −4 ± 2𝑖
𝑥 = −3 ± √14
These solutions will check.
I am not showing the work.
Solution: 𝒙 = −𝟒 ± 𝟐𝒊
These solutions will check.
I am not showing the work.
Solution: 𝒙 = −𝟑 ± √𝟏𝟒
57) x2 + 3x + 5 = 0
𝑥 2 + 3𝑥 = −5 ( I need to isolate the constant)
2
1
3 2
9
𝐶 = (2 ∗ 3) = (2) = 4
9
9
𝑥 2 + 3𝑥 + 4 = −5 + 4
to simplify the right side, do this:
9
5
9
−20
9
11
−5 + 4 = − 1 + 4 = 4 + 4 = − 4
3 2
(𝑥 + 2) = −
11
4
2
√(𝑥 + 3) = ±√−11
2
4
3
𝑥+2=±
𝑖√11
2
𝟑
Solution: 𝒙 = − 𝟐 ±
𝒊√𝟏𝟏
𝟐
=
−𝟑±𝒊√𝟏𝟏
𝟐
These solutions will check. I am not showing the
work.