TRY IT: Writing a Quadratic Function in Vertex Form

Choose the equation that shows a step in the process of completing the square on the given quadratic.

[tex]\( y = x^2 + 8x - 3 \)[/tex]

A. [tex]\( y = x^2 + 8x + 8 - 3 - 8 \)[/tex]
B. [tex]\( y = x^2 + 8x + 8 - 3 + 8 \)[/tex]
C. [tex]\( y = x^2 + 8x + 16 - 3 - 16 \)[/tex]
D. [tex]\( y = x^2 + 8x + 16 - 3 + 16 \)[/tex]

The vertex form of the function is:
[tex]\[ y = (x + \square)^2 + \square \][/tex]



Answer :

Let's transform the given quadratic function [tex]\( y = x^2 + 8x - 3 \)[/tex] into its vertex form by completing the square. Here's a step-by-step solution:

### Completing the Square Process:
1. Start with the original quadratic equation:
[tex]\[ y = x^2 + 8x - 3 \][/tex]

2. To complete the square, we focus on the quadratic and linear terms, [tex]\( x^2 + 8x \)[/tex].

3. Find the value needed to complete the square. This value is [tex]\( \left(\frac{8}{2}\right)^2 = 16 \)[/tex].

4. Add and subtract 16 inside the equation. Adding and subtracting the same number does not change the equality. Here's how it looks:
[tex]\[ y = x^2 + 8x + 16 - 16 - 3 \][/tex]
This matches the third option in the given choices:
[tex]\[ y = x^2 + 8x + 16 - 3 - 16 \][/tex]

5. Now, rewrite the equation by organizing the completed square. The equation now becomes:
[tex]\[ y = (x + 4)^2 - 19 \][/tex]

### Vertex Form of the Function:
The quadratic function in vertex form is:
[tex]\[ y = (x + 4)^2 - 19 \][/tex]

So, filling in the blanks, the vertex form of the function is:
[tex]\[ y = (x + \boxed{4})^2 + \boxed{-19} \][/tex]

Thus, the equation showing a step in the process of completing the square is:
[tex]\[ y = x^2 + 8x + 16 - 3 - 16 \][/tex]

And the vertex form of the quadratic function is:
[tex]\[ y = (x + 4)^2 - 19 \][/tex]