Solve (x+8)^2+x^2=256

$$( x + 8 ) ^ { 2 } + x ^ { 2 } = 256$$

Solve for x

$$x=4\sqrt{7}-4\approx 6.583005244$$

$$x=-4\sqrt{7}-4\approx -14.583005244$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$x^{2}+16x+64+x^{2}=256$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+16x+64=256$$

Subtract 256 from both sides.

$$2x^{2}+16x+64-256=0$$

Subtract 256 from 64 to get -192.

$$2x^{2}+16x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\left(-192\right)}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\left(-192\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\left(-192\right)}}{2\times 2}$$

Multiply -8 times -192.

$$x=\frac{-16±\sqrt{256+1536}}{2\times 2}$$

Add 256 to 1536.

$$x=\frac{-16±\sqrt{1792}}{2\times 2}$$

Take the square root of 1792.

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

Multiply 2 times 2.

$$x=\frac{-16±16\sqrt{7}}{4}$$

Now solve the equation x=\frac{-16±16\sqrt{7}}{4} when ± is plus. Add -16 to 16\sqrt{7}.

$$x=\frac{16\sqrt{7}-16}{4}$$

Divide -16+16\sqrt{7} by 4.

$$x=4\sqrt{7}-4$$

Now solve the equation x=\frac{-16±16\sqrt{7}}{4} when ± is minus. Subtract 16\sqrt{7} from -16.

$$x=\frac{-16\sqrt{7}-16}{4}$$

Divide -16-16\sqrt{7} by 4.

$$x=-4\sqrt{7}-4$$

The equation is now solved.

$$x=4\sqrt{7}-4 x=-4\sqrt{7}-4$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$x^{2}+16x+64+x^{2}=256$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$2x^{2}+16x+64=256$$

Subtract 64 from both sides.

$$2x^{2}+16x=256-64$$

Subtract 64 from 256 to get 192.

$$2x^{2}+16x=192$$

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=\frac{192}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=\frac{192}{2}$$

Divide 16 by 2.

$$x^{2}+8x=\frac{192}{2}$$

Divide 192 by 2.

$$x^{2}+8x=96$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=96+4^{2}$$

Square 4.

$$x^{2}+8x+16=96+16$$

Add 96 to 16.

$$x^{2}+8x+16=112$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=112$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{112}$$

Simplify.

$$x+4=4\sqrt{7} x+4=-4\sqrt{7}$$

Subtract 4 from both sides of the equation.

$$x=4\sqrt{7}-4 x=-4\sqrt{7}-4$$