Certainly! Let's express the quadratic expression [tex]\(x^2 - 8x + 5\)[/tex] in the form [tex]\((x - a)^2 - b\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers. This process is referred to as "completing the square."
### Step-by-Step Solution:
1. Start with the given quadratic expression:
[tex]\[
x^2 - 8x + 5
\][/tex]
2. Isolate the quadratic and linear terms:
We focus on [tex]\(x^2 - 8x\)[/tex] first. The goal is to transform this part into a perfect square trinomial.
3. Determine the term to complete the square:
To complete the square, take the coefficient of [tex]\(x\)[/tex], which is [tex]\(-8\)[/tex], and divide it by 2, then square the result:
[tex]\[
\left(\frac{-8}{2}\right)^2 = (-4)^2 = 16
\][/tex]
4. Add and subtract this square inside the expression:
Adding and subtracting 16 within the expression doesn't change its value. The expression becomes:
[tex]\[
x^2 - 8x + 16 - 16 + 5
\][/tex]
5. Group the perfect square trinomial:
Combine [tex]\(x^2 - 8x + 16\)[/tex] into a perfect square:
[tex]\[
(x - 4)^2 - 16 + 5
\][/tex]
6. Simplify the constants:
Combine the constants [tex]\(-16\)[/tex] and [tex]\(5\)[/tex] to simplify:
[tex]\[
(x - 4)^2 - 11
\][/tex]
7. Identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
By comparing [tex]\((x - a)^2 - b\)[/tex] to [tex]\((x - 4)^2 - 11\)[/tex], we can identify:
[tex]\[
a = 4 \quad \text{and} \quad b = 11
\][/tex]
Therefore, the expression [tex]\(x^2 - 8x + 5\)[/tex] can be written in the form [tex]\((x - 4)^2 - 11\)[/tex]. Hence, the integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = 4, \quad b = 11
\][/tex]
