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
