To find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that the equation
[tex]\[
x^2 - x + 5 = (x - a)^2 + b
\][/tex]
is satisfied, we will follow these steps:
1. Expand the right-hand side:
[tex]\[
(x - a)^2 + b = x^2 - 2ax + a^2 + b
\][/tex]
2. Set the left-hand side equal to the expanded right-hand side:
[tex]\[
x^2 - x + 5 = x^2 - 2ax + a^2 + b
\][/tex]
3. Compare the coefficients of the terms involving [tex]\( x \)[/tex] and the constant terms from both sides of the equation.
Let's start by comparing the coefficients of [tex]\( x^2 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] on the left-hand side is 1.
- The coefficient of [tex]\( x^2 \)[/tex] on the right-hand side is 1.
These coefficients are already equal, so no information about [tex]\( a \)[/tex] or [tex]\( b \)[/tex] can be derived from this.
Next, compare the coefficients of [tex]\( x \)[/tex]:
- The coefficient of [tex]\( x \)[/tex] on the left-hand side is [tex]\(-1\)[/tex].
- The coefficient of [tex]\( x \)[/tex] on the right-hand side is [tex]\(-2a\)[/tex].
Setting these equal gives:
[tex]\[
-1 = -2a \implies a = \frac{1}{2}
\][/tex]
Finally, compare the constant terms:
- The constant term on the left-hand side is 5.
- The constant terms on the right-hand side are [tex]\( a^2 + b \)[/tex].
Setting these equal gives:
[tex]\[
a^2 + b = 5
\][/tex]
Substitute [tex]\( a = \frac{1}{2} \)[/tex] into this equation:
[tex]\[
\left(\frac{1}{2}\right)^2 + b = 5 \implies \frac{1}{4} + b = 5 \implies b = 5 - \frac{1}{4} = \frac{20}{4} - \frac{1}{4} = \frac{19}{4}
\][/tex]
Thus, the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] that satisfy the equation are:
[tex]\[
a = \frac{1}{2}, \quad b = \frac{19}{4}
\][/tex]
