To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the equation
[tex]\[
x^2 - 4x + 9 = (x + a)^2 + b
\][/tex]
we will follow a step-by-step approach to equate coefficients from both sides of the equation.
### Step 1: Expand the Right-hand Side
First, expand [tex]\((x + a)^2 + b\)[/tex]:
[tex]\[
(x + a)^2 + b = (x^2 + 2ax + a^2) + b
\][/tex]
So the right-hand side becomes:
[tex]\[
x^2 + 2ax + a^2 + b
\][/tex]
### Step 2: Equate the Given Equation with the Expanded Right-hand Side
Rewrite the given equation by equating it with our expanded expression:
[tex]\[
x^2 - 4x + 9 = x^2 + 2ax + a^2 + b
\][/tex]
### Step 3: Compare Coefficients of Corresponding Powers of [tex]\(x\)[/tex]
Let's compare the coefficients of like terms from both sides of the equation.
1. Coefficient of [tex]\(x^2\)[/tex]:
[tex]\[
1 = 1
\][/tex]
2. Coefficient of [tex]\(x\)[/tex]:
[tex]\[
-4 = 2a
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
2a = -4 \implies a = -2
\][/tex]
3. Constant term:
[tex]\[
9 = a^2 + b
\][/tex]
Substitute [tex]\(a = -2\)[/tex] into this equation:
[tex]\[
9 = (-2)^2 + b
\][/tex]
Simplify and solve for [tex]\(b\)[/tex]:
[tex]\[
9 = 4 + b \implies b = 9 - 4 \implies b = 5
\][/tex]
### Step 4: State the Solution
The values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the equation are:
[tex]\[
a = -2 \quad \text{and} \quad b = 5
\][/tex]
