To find the value of [tex]\( g \)[/tex] that makes the equation [tex]\( (x+7)(x-4) = x^2 + 9x - 28 \)[/tex] true, let's follow a step-by-step approach to solve the problem:
1. Expand the left-hand side of the equation:
Let's expand [tex]\( (x+7)(x-4) \)[/tex] using the distributive property (also known as the FOIL method for binomials):
[tex]\[
(x+7)(x-4) = x(x) + x(-4) + 7(x) + 7(-4)
\][/tex]
2. Simplify the expression from the expansion:
Calculate each individual term:
[tex]\[
(x+7)(x-4) = x^2 - 4x + 7x - 28
\][/tex]
Combine like terms:
[tex]\[
x^2 - 4x + 7x - 28 = x^2 + 3x - 28
\][/tex]
So, the left-hand side of the equation [tex]\( (x+7)(x-4) \)[/tex] simplifies to [tex]\( x^2 + 3x - 28 \)[/tex].
3. Compare the simplified left-hand side with the right-hand side:
The given equation is:
[tex]\[
(x+7)(x-4) = x^2 + 9x - 28
\][/tex]
Substituting the simplified left-hand side, we get:
[tex]\[
x^2 + 3x - 28 = x^2 + 9x - 28
\][/tex]
4. Set up the equation to find [tex]\( g \)[/tex]:
Since the [tex]\( x^2 \)[/tex] and the constant term [tex]\(-28\)[/tex] are the same on both sides, we only need to compare the coefficients of the [tex]\( x \)[/tex] term. Therefore, we have:
[tex]\[
3x = 9x
\][/tex]
To find the value of [tex]\( g \)[/tex], which is the coefficient of [tex]\( x \)[/tex] on the right-hand side, we set:
[tex]\[
3 = 9
\][/tex]
5. Determine the value of [tex]\( g \)[/tex]:
From the equation above, we see that the coefficient of [tex]\( x \)[/tex] from the left-hand side is [tex]\( 3 \)[/tex], and on the right-hand side, it is given as [tex]\( 9 \)[/tex]. Therefore, [tex]\( g \)[/tex] must be equal to [tex]\( 9 \)[/tex].
So, the value of [tex]\( g \)[/tex] that makes the equation true is:
[tex]\[
g = 3
\][/tex]
