Question

complete the square to re-write the quadratic function in vertex form:
$y = x^2 + 10x - 8$

Explanation:

Step1: Isolate the x-terms

We start with the quadratic function \( y = x^2 + 10x - 8 \). First, we isolate the terms with \( x \) by moving the constant term to the other side (or keeping it separate for now). So we can rewrite it as \( y + 8 = x^2 + 10x \).

Step2: Complete the square for the x-terms

To complete the square for the expression \( x^2 + 10x \), we take half of the coefficient of \( x \), which is \( \frac{10}{2}=5 \), and then square it: \( 5^2 = 25 \). We add this value to both sides of the equation. So we have \( y + 8+ 25= x^2 + 10x + 25 \).

Step3: Rewrite as a perfect square and simplify

The right - hand side \( x^2 + 10x + 25 \) can be written as \( (x + 5)^2 \) (since \( (a + b)^2=a^2 + 2ab + b^2 \), here \( a = x \) and \( b = 5 \), and \( 2ab=10x \)). The left - hand side simplifies to \( y+33 \). So the equation becomes \( y+33=(x + 5)^2 \).

Step4: Solve for y

Now, we solve for \( y \) by subtracting 33 from both sides. We get \( y=(x + 5)^2-33 \).

Answer:

\( y=(x + 5)^2-33 \)