Certainly! We'll go through the process of completing the square for the given polynomial step-by-step:
Given:
[tex]\[ f(x) = 4 + 3x - x^2 \][/tex]
1. Rewriting the polynomial in standard form:
[tex]\[ f(x) = -x^2 + 3x + 4 \][/tex]
2. Factor out [tex]\(-1\)[/tex] from the quadratic term:
[tex]\[ f(x) = -1(x^2 - 3x) + 4 \][/tex]
3. Complete the square inside the parentheses:
We need to complete the square for the expression [tex]\( x^2 - 3x \)[/tex]. To do this, add and subtract [tex]\((\frac{3}{2})^2\)[/tex] inside the parentheses:
[tex]\[ x^2 - 3x = \left(x - \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 \][/tex]
4. Substitute this back into the polynomial:
[tex]\[ f(x) = -1\left[\left(x - \frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2\right] + 4 \][/tex]
[tex]\[ f(x) = -1\left(x - \frac{3}{2}\right)^2 + \frac{9}{4} + 4 \][/tex]
5. Simplify the expression:
By simplifying the constants:
[tex]\[ 4 = \frac{16}{4} \][/tex]
[tex]\[ \frac{9}{4} + \frac{16}{4} = \frac{25}{4} \][/tex]
So,
[tex]\[ f(x) = -1\left(x - \frac{3}{2}\right)^2 + \frac{25}{4} \][/tex]
6. Identifying the values of [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex]:
By comparing the expression [tex]\[ f(x) = -1\left(x - \frac{3}{2}\right)^2 + \frac{25}{4} \][/tex] to the form [tex]\[ f(x) = P - Q\left(C x + R\right)^2 \][/tex], we get:
[tex]\[
\begin{align*}
P & = \frac{25}{4} \\
Q & = 1 \\
R & = -\frac{3}{2}
\end{align*}
\][/tex]
Therefore, the values of [tex]\( P \)[/tex], [tex]\( Q \)[/tex], and [tex]\( R \)[/tex] are [tex]\( 6.25 \)[/tex], [tex]\( 1 \)[/tex], and [tex]\( -1.5 \)[/tex] respectively.
