Answer:
D. completing the square
Explanation:
The process shown rearranges the standard-form quadratic equation to vertex form. It involves factoring out the leading coefficient and distributing the constant so that the variable terms can be written as a perfect square.
What is a perfect square trinomial?
A perfect square trinomial is the square of a binomial. It has the form ...
(x +b)² = x² +2bx +b²
Given two variable terms, x² and 2bx, the perfect square trinomial can be formed by adding b², which is the square of half the x-term coefficient.
When b² is added to the sum (x² +2bx), the square is complete. The sum (x² +2bx +b²) can be written as the square (x +b)².
Completing the square
This process of adding b² to the sum (x² +2bx) will change the polynomial unless a similar quantity is subtracted. That is why the 3rd line of the problem statement show 4 being added and subtracted inside parentheses:
y = 5(x² +4x +4 -4) -17
When we're applying this transformation, we do not want to change the equation. We simply want to rearrange it.
In the next step, the subtracted term is brought outside the parentheses. This lets us write the quantity in parentheses as a perfect square.
y = 5(x² +4x +4) +5(-4) -17
y = 5(x² +4x +4) -20 -17
This process of adding and subtracting a suitable quantity and rearranging the equation so the variable terms make a perfect square is called ...
"completing the square."
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Additional comment
The equation in this form is said to be in "vertex form."
y = a(x -h)² +k
In this form, the constant 'a' is a vertical scale factor; the ordered pair (h, k) identifies the vertex of the quadratic on a graph.