Question

choose the equation that shows a step in the process of completing the square on the given quadratic. y = x² + 8x - 3 y = x² + 8x + 8 - 3 - 8 y = x² + 8x + 8 - 3 + 8 y = x² + 8x + 16 - 3 - 16 y = x² + 8x + 16 - 3 + 16 complete the vertex form of the function is y = (x + \square)² + \square. done

Explanation:

Step1: Recall Completing the Square

For a quadratic \(y = x^2 + bx + c\), to complete the square, we take half of \(b\), square it, add and subtract that value. Here, \(b = 8\), half of 8 is 4, and \(4^2 = 16\).

Step2: Apply to the Quadratic

Given \(y = x^2 + 8x - 3\), we add and subtract 16: \(y = x^2 + 8x + 16 - 3 - 16\) (because we added 16, we must subtract 16 to keep the equation equivalent).

Step3: Find Vertex Form

From \(y = x^2 + 8x + 16 - 3 - 16\), we can rewrite \(x^2 + 8x + 16\) as \((x + 4)^2\). Then simplify the constants: \(-3 - 16 = -19\). So the vertex form is \(y=(x + 4)^2 - 19\).

Answer:

For the equation step: \(y = x^2 + 8x + 16 - 3 - 16\) (the third option).
For the vertex form: The first box is \(4\), the second box is \(-19\), so \(y=(x + \boldsymbol{4})^2 + \boldsymbol{(-19)}\) (or \(y=(x + 4)^2 - 19\)).