Question

What is 2x^2+9x-18 in vertex form?

Answer 1

Complete the square:

2x² + 9x - 18

2(x² + 9/2 x) - 18

2(x² + 9/2 x + 81/16 - 81/16) - 18

2(x² + 2 (9/4) x + (9/4)²) - 81/8 - 18

2(x + 9/4)² - 225/8

If you're not familiar with the method, the idea is to get a quadratic from the standard form

ax² + bx + c

into its vertex form

a (x - h)² + k

where (h, k) is the vertex of the parabola associated with the quadratic.

Expand the vertex form to get

a (x² - 2xh + h²) + k

ax² - 2ahx + ah² + k

For the two forms to be equivalent, we must have

-2ah = b

ah² + k = c

which means

h = -b/(2a)

k = c - ah² = c - b²/(4a)

In this case, a = 2, b = 9, and c = -18, so

h = -9/(2 • 2) = -9/4

k = 18 - 9²/(4 • 2) = -225/8