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]
