Solve 2x=324-x^2

$$2 x = 324 - x ^ { 2 }$$

Solve for x

$$x=5\sqrt{13}-1\approx 17.027756377$$

$$x=-5\sqrt{13}-1\approx -19.027756377$$

Steps Using the Quadratic Formula

Subtract 324 from both sides.

$$2x-324=-x^{2}$$

Add x^{2} to both sides.

$$2x-324+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+2x-324=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -324 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-324\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-324\right)}}{2}$$

Multiply -4 times -324.

$$x=\frac{-2±\sqrt{4+1296}}{2}$$

Add 4 to 1296.

$$x=\frac{-2±\sqrt{1300}}{2}$$

Take the square root of 1300.

$$x=\frac{-2±10\sqrt{13}}{2}$$

Now solve the equation x=\frac{-2±10\sqrt{13}}{2} when ± is plus. Add -2 to 10\sqrt{13}.

$$x=\frac{10\sqrt{13}-2}{2}$$

Divide -2+10\sqrt{13} by 2.

$$x=5\sqrt{13}-1$$

Now solve the equation x=\frac{-2±10\sqrt{13}}{2} when ± is minus. Subtract 10\sqrt{13} from -2.

$$x=\frac{-10\sqrt{13}-2}{2}$$

Divide -2-10\sqrt{13} by 2.

$$x=-5\sqrt{13}-1$$

The equation is now solved.

$$x=5\sqrt{13}-1 x=-5\sqrt{13}-1$$

Steps for Completing the Square

Add x^{2} to both sides.

$$2x+x^{2}=324$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+2x=324$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=324+1^{2}$$

Square 1.

$$x^{2}+2x+1=324+1$$

Add 324 to 1.

$$x^{2}+2x+1=325$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=325$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{325}$$

Simplify.

$$x+1=5\sqrt{13} x+1=-5\sqrt{13}$$

Subtract 1 from both sides of the equation.

$$x=5\sqrt{13}-1 x=-5\sqrt{13}-1$$