Sure, let me walk you through the steps needed to rewrite the equation in vertex form:
The given equation is y = -x^2 + 6x - 5. We are tasked with transforming this standard form of a quadratic equation into vertex form.
Step 1: Group the x-terms
We rewrite the equation as follows: y = -(x^2 - 6x) - 5
Step 2: Complete the square
To do this, we need an equation in the format x^2 - bx. Once we have that, we can complete the square by adding (b/2)^2.
In our case, b = -6 (from the x-terms in our grouped equation).
Next, we calculate b/2 and square it. This yields (-6/2)^2 = 9.
Step 3: Keep the equation balanced
In order to add inside the parenthesis to complete the square, we must also subtract the same value from outside to keep the equation balanced.
Thus, we end up with the equation: y = -(x^2 - 6x + 9) - 5 + 9.
Step 4: Simplify
Finally, we simplify the equation to get it in vertex form. We rewrite it as follows: y = -(x - 3)^2 + 4.
So, the vertex form of the given quadratic equation is y = -(x - 3)^2 + 4.