Solve x^2+64x+576=0

$${ x }^{ 2 } +64x+576=0$$

Solve for x

$$x=8\sqrt{7}-32\approx -10.833989511$$

$$x=-8\sqrt{7}-32\approx -53.166010489$$

Steps Using the Quadratic Formula

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.

$$x^{2}+64x+576=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 64 for b, and 576 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-64±\sqrt{64^{2}-4\times 576}}{2}$$

Square 64.

$$x=\frac{-64±\sqrt{4096-4\times 576}}{2}$$

Multiply -4 times 576.

$$x=\frac{-64±\sqrt{4096-2304}}{2}$$

Add 4096 to -2304.

$$x=\frac{-64±\sqrt{1792}}{2}$$

Take the square root of 1792.

$$x=\frac{-64±16\sqrt{7}}{2}$$

Now solve the equation x=\frac{-64±16\sqrt{7}}{2} when ± is plus. Add -64 to 16\sqrt{7}.

$$x=\frac{16\sqrt{7}-64}{2}$$

Divide -64+16\sqrt{7} by 2.

$$x=8\sqrt{7}-32$$

Now solve the equation x=\frac{-64±16\sqrt{7}}{2} when ± is minus. Subtract 16\sqrt{7} from -64.

$$x=\frac{-16\sqrt{7}-64}{2}$$

Divide -64-16\sqrt{7} by 2.

$$x=-8\sqrt{7}-32$$

The equation is now solved.

$$x=8\sqrt{7}-32 x=-8\sqrt{7}-32$$

Steps for Completing the Square

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.

$$x^{2}+64x+576=0$$

Subtract 576 from both sides of the equation.

$$x^{2}+64x+576-576=-576$$

Subtracting 576 from itself leaves 0.

$$x^{2}+64x=-576$$

Divide 64, the coefficient of the x term, by 2 to get 32. Then add the square of 32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+64x+32^{2}=-576+32^{2}$$

Square 32.

$$x^{2}+64x+1024=-576+1024$$

Add -576 to 1024.

$$x^{2}+64x+1024=448$$

Factor x^{2}+64x+1024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+32\right)^{2}=448$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+32\right)^{2}}=\sqrt{448}$$

Simplify.

$$x+32=8\sqrt{7} x+32=-8\sqrt{7}$$

Subtract 32 from both sides of the equation.

$$x=8\sqrt{7}-32 x=-8\sqrt{7}-32$$