When rewriting the function [tex]y = x^2 - 5x - 20[/tex] by completing the square, at which of the following equations will you arrive?

A. [tex]y = \left(x - \frac{5}{2}\right)^2 - \frac{55}{4}[/tex]
B. [tex]y = \left(x + \frac{5}{2}\right)^2 + \frac{55}{4}[/tex]
C. [tex]y = \left(x + \frac{5}{2}\right)^2 + \frac{105}{4}[/tex]
D. [tex]y = \left(x - \frac{5}{2}\right)^2 - \frac{105}{4}[/tex]



Answer :

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]