To express the quadratic equation y = x^2 + 18x + 77 in the form y = (x + a)^2 - b, you need to complete the square. Here's how you can do it:
1. Start with the equation:
y = x^2 + 18x + 77
2. To complete the square, focus on the x^2 and x terms. You'll want to add and subtract a value that makes the x^2 term a perfect square trinomial. In this case, that value is (18/2)^2 = 9^2 = 81.
3. Rewrite the equation by adding and subtracting 81:
y = x^2 + 18x + 81 - 81 + 77
4. Now, group the perfect square trinomial and the constants:
y = (x^2 + 18x + 81) - 81 + 77
5. The grouped part is now a perfect square trinomial:
y = (x + 9)^2 - 81 + 77
6. Simplify the constants:
y = (x + 9)^2 - 4
So, the quadratic equation y = x^2 + 18x + 77 can be expressed in the form y = (x + 9)^2 - 4.