To solve for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] in the given equation [tex]\( x^2 + 8x + 10 = (x + p)^2 + q \)[/tex], we will proceed step by step.
1. Step 1: Expand [tex]\( (x + p)^2 \)[/tex]
Start by expanding the expression on the right-hand side of the equation:
[tex]\[
(x + p)^2 = x^2 + 2px + p^2
\][/tex]
Therefore, we can rewrite the right-hand side of the equation as:
[tex]\[
(x + p)^2 + q = x^2 + 2px + p^2 + q
\][/tex]
2. Step 2: Set the expanded form equal to the original equation
Now, we equate the left-hand side [tex]\( x^2 + 8x + 10 \)[/tex] with the expanded form [tex]\( x^2 + 2px + p^2 + q \)[/tex]:
[tex]\[
x^2 + 8x + 10 = x^2 + 2px + p^2 + q
\][/tex]
3. Step 3: Compare coefficients
To solve for [tex]\( p \)[/tex] and [tex]\( q \)[/tex], compare the coefficients of [tex]\( x \)[/tex] and the constant terms on both sides.
- Coefficient of [tex]\( x^2 \)[/tex]:
On both sides, the coefficient is 1, so this term cancels out.
- Coefficient of [tex]\( x \)[/tex]:
[tex]\[
8 = 2p
\][/tex]
Solve for [tex]\( p \)[/tex]:
[tex]\[
p = \frac{8}{2} = 4
\][/tex]
- Constant terms:
[tex]\[
10 = p^2 + q
\][/tex]
Substitute [tex]\( p = 4 \)[/tex] into this equation:
[tex]\[
10 = 4^2 + q
\][/tex]
Which simplifies to:
[tex]\[
10 = 16 + q
\][/tex]
Solving for [tex]\( q \)[/tex]:
[tex]\[
q = 10 - 16
\][/tex]
Therefore:
[tex]\[
q = -6
\][/tex]
4. Step 4: Conclusion
We have found the values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]:
[tex]\[
p = 4
\][/tex]
[tex]\[
q = -6
\][/tex]
Thus, the solution is:
[tex]\( p = 4 \)[/tex] and [tex]\( q = -6 \)[/tex].
