To find the value of [tex]\( x \)[/tex] that makes the equation true, we will solve the equation step by step.
Given equation:
[tex]\[
(x + 7)(x - 4) = x^2 + 9x - 29
\][/tex]
Step 1: Expand the left side of the equation.
[tex]\[
(x + 7)(x - 4)
\][/tex]
Expanding this using the distributive property (FOIL method), we get:
[tex]\[
x \cdot x + x \cdot (-4) + 7 \cdot x + 7 \cdot (-4)
\][/tex]
[tex]\[
= x^2 - 4x + 7x - 28
\][/tex]
Combine like terms:
[tex]\[
x^2 + 3x - 28
\][/tex]
Now the equation looks like this:
[tex]\[
x^2 + 3x - 28 = x^2 + 9x - 29
\][/tex]
Step 2: Subtract [tex]\( x^2 + 9x - 29 \)[/tex] from both sides of the equation to set it to 0:
[tex]\[
(x^2 + 3x - 28) - (x^2 + 9x - 29) = 0
\][/tex]
Simplify the equation:
[tex]\[
x^2 + 3x - 28 - x^2 - 9x + 29 = 0
\][/tex]
Combine like terms:
[tex]\[
3x - 9x - 28 + 29 = 0
\][/tex]
[tex]\[
-6x + 1 = 0
\][/tex]
Step 3: Solve for [tex]\( x \)[/tex]:
[tex]\[
-6x + 1 = 0
\][/tex]
Subtract 1 from both sides:
[tex]\[
-6x = -1
\][/tex]
Divide both sides by -6:
[tex]\[
x = \frac{1}{6}
\][/tex]
Therefore, the value of [tex]\( g \)[/tex] that makes the equation true is:
[tex]\[
\frac{1}{6}
\][/tex]
This value is not provided in the given options of -11, -3, 3, or 11. None of the provided options satisfy the equation, since the correct solution is [tex]\( \frac{1}{6} \)[/tex].
