To find the values of [tex]\( b \)[/tex] that satisfy the equation [tex]\( 4(3b + 2)^2 = 64 \)[/tex], we can proceed with the following steps:
1. Divide both sides by 4:
[tex]\[
4(3b + 2)^2 = 64
\][/tex]
Divide both sides by 4:
[tex]\[
(3b + 2)^2 = 16
\][/tex]
2. Take the square root of both sides:
[tex]\[
\sqrt{(3b + 2)^2} = \pm \sqrt{16}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex]:
[tex]\[
3b + 2 = 4 \quad \text{or} \quad 3b + 2 = -4
\][/tex]
3. Solve each equation for [tex]\( b \)[/tex]:
- For [tex]\( 3b + 2 = 4 \)[/tex]:
[tex]\[
3b + 2 = 4
\][/tex]
Subtract 2 from both sides:
[tex]\[
3b = 2
\][/tex]
Divide by 3:
[tex]\[
b = \frac{2}{3}
\][/tex]
- For [tex]\( 3b + 2 = -4 \)[/tex]:
[tex]\[
3b + 2 = -4
\][/tex]
Subtract 2 from both sides:
[tex]\[
3b = -6
\][/tex]
Divide by 3:
[tex]\[
b = -2
\][/tex]
Therefore, the values of [tex]\( b \)[/tex] that satisfy the equation [tex]\( 4(3b + 2)^2 = 64 \)[/tex] are:
[tex]\[
b = \frac{2}{3} \quad \text{and} \quad b = -2
\][/tex]
Among the given options:
1. [tex]\( b = \frac{2}{3} \)[/tex] and [tex]\( b = -2 \)[/tex] [tex]\( \quad \text{(correct)} \)[/tex]
2. [tex]\( b = 2 \)[/tex] and [tex]\( b = \frac{10}{3} \)[/tex]
3. [tex]\( b = \frac{2}{3} \)[/tex] and [tex]\( b = 3 \)[/tex]
4. [tex]\( b = 2 \)[/tex] and [tex]\( b = \frac{-10}{3} \)[/tex]
Thus, the correct answer is:
[tex]\[
b = \frac{2}{3} \quad \text{and} \quad b = -2
\][/tex]
