Question

Let $x$ be the positive number such that $2x^2 = 4x + 9.$

If $x$ can be written in simplified form as $\dfrac{a + \sqrt{b}}{c}$ such that $a,$ $b,$ and $c$ are positive integers, what is $a + b + c$?

Answer 1

Solve for x. I'll do it by completing the square:

2x² = 4x + 9

2x² - 4x = 9

x² - 2x = 9/2

x² - 2x + 1 = 11/2

(x - 1)² = 11/2

x - 1 = ± √(11/2)

x = 1 ± √(11/2)

x = 1 ± √22/2

x = (2 ± √22)/2

Take the solution with the positive square root. Then it follows that a = c = 2 and b = 22. So a + b + c = 26.