Final answer:
The quadratic equation y=9x^2+162x+725 is converted to vertex form by completing the square, resulting in y=9(x+9)^2-4 with the vertex at (-9, -4).
Step-by-step explanation:
To rewrite the quadratic function y=9x^2+162x+725 in vertex form by completing the square, follow these steps:
First, factor the coefficient of the x^2 term out of the first two terms: y = 9(x^2 + 18x) + 725.
Next, to complete the square for the expression x^2 + 18x, find (1/2)*18 which is 9, and then square it, getting 81.
Add and subtract this value inside the parentheses to complete the square: y = 9(x^2 + 18x + 81 - 81) + 725.
Simplify inside the parentheses to form a perfect square: y = 9((x + 9)^2 - 81) + 725.
Distribute the 9 and combine like terms to get the vertex form: y = 9(x + 9)^2 - 729 + 725, which simplifies to y = 9(x + 9)^2 - 4.
The quadratic function in vertex form is y = 9(x + 9)^2 - 4, where the vertex of the parabola is (-9, -4).