To determine what value of [tex]\( b \)[/tex] causes Equation A and Equation B to have the same solutions, follow these steps:
1. Solve Equation B:
[tex]\((x + 3)^2 = 64\)[/tex]
To find the solutions for [tex]\( x \)[/tex], we take the square root of both sides first:
[tex]\[
x + 3 = \pm \sqrt{64}
\][/tex]
[tex]\[
x + 3 = \pm 8
\][/tex]
This results in two separate equations:
[tex]\[
x + 3 = 8 \quad \text{or} \quad x + 3 = -8
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
For [tex]\( x + 3 = 8 \)[/tex]:
[tex]\[
x = 8 - 3 = 5
\][/tex]
For [tex]\( x + 3 = -8 \)[/tex]:
[tex]\[
x = -8 - 3 = -11
\][/tex]
So, the solutions for Equation B are [tex]\( x = 5 \)[/tex] and [tex]\( x = -11 \)[/tex].
3. Substitute [tex]\( x \)[/tex] values from Equation B into Equation A:
Equation A: [tex]\( x^2 + bx - 8 = 47 \)[/tex]
For [tex]\( x = 5 \)[/tex]:
[tex]\[
5^2 + b(5) - 8 = 47
\][/tex]
Simplify:
[tex]\[
25 + 5b - 8 = 47
\][/tex]
[tex]\[
17 + 5b = 47
\][/tex]
Subtract 17 from both sides:
[tex]\[
5b = 30
\][/tex]
Divide by 5:
[tex]\[
b = 6
\][/tex]
For [tex]\( x = -11 \)[/tex]:
[tex]\[
(-11)^2 + b(-11) - 8 = 47
\][/tex]
Simplify:
[tex]\[
121 - 11b - 8 = 47
\][/tex]
[tex]\[
113 - 11b = 47
\][/tex]
Subtract 113 from both sides:
[tex]\[
-11b = -66
\][/tex]
Divide by -11:
[tex]\[
b = 6
\][/tex]
Therefore, the value of [tex]\( b \)[/tex] that causes Equation A and Equation B to have the same solutions is [tex]\( \boxed{6} \)[/tex].
