You need to solve the following equation:
-0.5x^2 +36x -161 = 451
Set right side to 0
-0.5x^2 +36x -612 = 0
Multiply by -2 to get rid of decimal and leading neg coefficient.
x^2 -72x + 1224 = 0
Use quadratic formula:
a = 1, b = -72, c = 1224
[tex]x = \frac{-(-72) \pm \sqrt{(-72)^2 -4(1)(1224)}}{2(1)} \\ \\ x = \frac{72 \pm \sqrt{288}}{2} \\ \\ x = 27.51, 44.49[/tex]
Therefore the min price to have 451 profit is $27.51
