To rewrite the function [tex]\( y = x^2 - 5x - 20 \)[/tex] by completing the square, we follow these detailed steps:
1. Identify the coefficient of [tex]\( x \)[/tex]: In this quadratic equation, the coefficient of [tex]\( x \)[/tex] is -5.
2. Complete the square:
- Begin with the equation [tex]\( y = x^2 - 5x - 20 \)[/tex].
- We want to form a perfect square trinomial from the [tex]\( x \)[/tex]-terms. To do this, we take half of the [tex]\( x \)[/tex]-coefficient (which is [tex]\(-5\)[/tex]), divide it by 2, and then square it:
[tex]\[
\left(\frac{-5}{2}\right)^2 = \left(-\frac{5}{2}\right)^2 = \frac{25}{4}
\][/tex]
- Add and subtract [tex]\(\frac{25}{4}\)[/tex] inside the equation:
[tex]\[
y = x^2 - 5x + \frac{25}{4} - \frac{25}{4} - 20
\][/tex]
3. Regroup the terms to complete the square:
- Rewrite the quadratic portion as a perfect square:
[tex]\[
y = \left( x^2 - 5x + \frac{25}{4} \right) - \frac{25}{4} - 20
\][/tex]
- Factor the perfect square trinomial:
[tex]\[
y = \left( x - \frac{5}{2} \right)^2 - \frac{25}{4} - 20
\][/tex]
4. Simplify the constants:
- Combine [tex]\(\frac{25}{4}\)[/tex] and -20. To do this, rewrite -20 as a fraction with the same denominator:
[tex]\[
20 = \frac{80}{4}
\][/tex]
- Therefore:
[tex]\[
y = \left( x - \frac{5}{2} \right)^2 - \frac{25}{4} - \frac{80}{4}
\][/tex]
- Combine the fractions:
[tex]\[
y = \left( x - \frac{5}{2} \right)^2 - \frac{105}{4}
\][/tex]
Thus, the equation we arrive at after completing the square is:
[tex]\[
y = \left( x - \frac{5}{2} \right)^2 - \frac{105}{4}
\][/tex]
So, the correct choice is:
[tex]\[ y = \left(x - \frac{5}{2}\right)^2 - \frac{105}{4} \][/tex]
