Certainly! Let's solve for the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] by matching the coefficients of the given equation:
[tex]\[ x^2 + 8x - 13 = (x + p)^2 + q \][/tex]
First, let's expand the right-hand side of the equation:
[tex]\[ (x + p)^2 + q = x^2 + 2px + p^2 + q \][/tex]
Now, we match the corresponding coefficients of [tex]\( x^2 \)[/tex], [tex]\( x \)[/tex], and the constant terms from both sides of the equation.
### Step 1: Match the coefficient of [tex]\( x^2 \)[/tex]
From the left-hand side:
[tex]\[ x^2 + 8x - 13 \][/tex]
From the right-hand side:
[tex]\[ x^2 + 2px + p^2 + q \][/tex]
We see that the coefficients of [tex]\( x^2 \)[/tex] are already equal:
[tex]\[ 1 = 1 \][/tex]
### Step 2: Match the coefficient of [tex]\( x \)[/tex]
From the left-hand side:
[tex]\[ 8x \][/tex]
From the right-hand side:
[tex]\[ 2px \][/tex]
Set the coefficients of [tex]\( x \)[/tex] equal to each other:
[tex]\[ 8 = 2p \][/tex]
Solve for [tex]\( p \)[/tex]:
[tex]\[
2p = 8 \\
p = \frac{8}{2} \\
p = 4
\][/tex]
### Step 3: Match the constant terms
From the left-hand side:
[tex]\[ -13 \][/tex]
From the right-hand side:
[tex]\[ p^2 + q \][/tex]
Set the constant terms equal to each other, substitute [tex]\( p = 4 \)[/tex]:
[tex]\[ -13 = 4^2 + q \][/tex]
Solve for [tex]\( q \)[/tex]:
[tex]\[
-13 = 16 + q \\
q = -13 - 16 \\
q = -29
\][/tex]
Thus, the values are:
[tex]\[ p = 4 \][/tex]
[tex]\[ q = -29 \][/tex]
