To complete the square and rewrite the quadratic function in vertex form, we need to follow these steps:
Step 1: Group the x terms and the constant term separately:
y = x^2 - 5x - 7
y = (x^2 - 5x) - 7
Step 2: Complete the square by adding and subtracting the square of half the coefficient of x from the x terms:
y = (x^2 - 5x + (5/2)^2) - (5/2)^2 - 7
Step 3: Simplify the expression inside the parentheses:
y = (x - (5/2))^2 - (25/4) - 7
Step 4: Combine the constant terms:
y = (x - (5/2))^2 - (25/4) - (28/4)
Step 5: Simplify the constant terms:
y = (x - (5/2))^2 - (53/4)
Therefore, the quadratic function in vertex form is: y = (x - (5/2))^2 - (53/4).