To rewrite the quadratic expression [tex]\( x^2 + 10x - 7 \)[/tex] in the form [tex]\( (x + a)^2 + b \)[/tex], we need to complete the square. Here's a detailed, step-by-step process to achieve this:
1. Start with the given quadratic expression:
[tex]\[
x^2 + 10x - 7
\][/tex]
2. Focus on the quadratic and linear terms ([tex]\( x^2 + 10x \)[/tex]). To complete the square, we want to form a perfect square trinomial from these terms.
3. Determine the number needed to complete the square:
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\( 10 \)[/tex], and divide it by 2, then square the result:
[tex]\[
\left( \frac{10}{2} \right)^2 = 5^2 = 25
\][/tex]
4. Rewrite the quadratic and linear terms as a perfect square trinomial:
[tex]\[
x^2 + 10x + 25
\][/tex]
- We added [tex]\( 25 \)[/tex], so we must also subtract [tex]\( 25 \)[/tex] to maintain the equality. Thus,
[tex]\[
x^2 + 10x = (x + 5)^2 - 25
\][/tex]
5. Substitute back into the original expression:
[tex]\[
x^2 + 10x - 7 = (x + 5)^2 - 25 - 7
\][/tex]
6. Combine the constants:
[tex]\[
-25 - 7 = -32
\][/tex]
7. Write the final expression:
[tex]\[
x^2 + 10x - 7 = (x + 5)^2 - 32
\][/tex]
Thus, comparing the expression [tex]\( (x + a)^2 + b \)[/tex] with our final result [tex]\( (x + 5)^2 - 32 \)[/tex], we find:
[tex]\[
a = 5 \quad \text{and} \quad b = -32
\][/tex]
Therefore, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are:
[tex]\[
a = 5 \quad \text{and} \quad b = -32
\][/tex]
