To rewrite the function [tex]\( f(x) = x^2 - 9x + 14 \)[/tex] by completing the square, we will follow these steps:
1. Start with the original quadratic function:
[tex]\[
f(x) = x^2 - 9x + 14
\][/tex]
2. Identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic expression [tex]\( ax^2 + bx + c \)[/tex]:
[tex]\[
a = 1, \quad b = -9, \quad c = 14
\][/tex]
3. Complete the square:
- Take half of the coefficient of [tex]\( x \)[/tex], which is [tex]\( b \)[/tex]. In this case, half of [tex]\( -9 \)[/tex] is [tex]\( -4.5 \)[/tex]:
[tex]\[
\frac{b}{2} = -\frac{9}{2} = -4.5
\][/tex]
- Square this value:
[tex]\[
\left(-\frac{9}{2}\right)^2 = \left(-4.5\right)^2 = 20.25
\][/tex]
- Rewrite the original function as the square of a binomial and adjust the constant term:
[tex]\[
f(x) = x^2 - 9x + 14
\][/tex]
[tex]\[
f(x) = \left(x - 4.5\right)^2 - 20.25 + 14
\][/tex]
- Combine the constant terms:
[tex]\[
f(x) = \left(x - 4.5\right)^2 - 6.25
\][/tex]
4. Write the completed square form:
[tex]\[
f(x) = \left(x - 4.5\right)^2 - 6.25
\][/tex]
Thus, by completing the square, the function [tex]\( f(x) = x^2 - 9x + 14 \)[/tex] can be rewritten as:
[tex]\[
f(x) = \left( x - 4.5 \right)^2 - 6.25
\][/tex]
In the form given in the problem:
[tex]\[
f(x) = \boxed{1}(x + \boxed{4.5})^2 + \boxed{-6.25}
\][/tex]
