Solve x^2+(x+8)^2=12^2

$$x ^ { 2 } + ( x + 8 ) ^ { 2 } = 12 ^ { 2 }$$

Solve for x

$$x=2\sqrt{14}-4\approx 3.483314774$$

$$x=-2\sqrt{14}-4\approx -11.483314774$$

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}.

$$x^{2}+x^{2}+16x+64=12^{2}$$

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

$$2x^{2}+16x+64=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$2x^{2}+16x+64=144$$

Subtract 144 from both sides.

$$2x^{2}+16x+64-144=0$$

Subtract 144 from 64 to get -80.

$$2x^{2}+16x-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 16 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\left(-80\right)}}{2\times 2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 2\left(-80\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-16±\sqrt{256-8\left(-80\right)}}{2\times 2}$$

Multiply -8 times -80.

$$x=\frac{-16±\sqrt{256+640}}{2\times 2}$$

Add 256 to 640.

$$x=\frac{-16±\sqrt{896}}{2\times 2}$$

Take the square root of 896.

$$x=\frac{-16±8\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-16±8\sqrt{14}}{4}$$

Now solve the equation x=\frac{-16±8\sqrt{14}}{4} when ± is plus. Add -16 to 8\sqrt{14}.

$$x=\frac{8\sqrt{14}-16}{4}$$

Divide -16+8\sqrt{14} by 4.

$$x=2\sqrt{14}-4$$

Now solve the equation x=\frac{-16±8\sqrt{14}}{4} when ± is minus. Subtract 8\sqrt{14} from -16.

$$x=\frac{-8\sqrt{14}-16}{4}$$

Divide -16-8\sqrt{14} by 4.

$$x=-2\sqrt{14}-4$$

The equation is now solved.

$$x=2\sqrt{14}-4 x=-2\sqrt{14}-4$$

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}.

$$x^{2}+x^{2}+16x+64=12^{2}$$

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

$$2x^{2}+16x+64=12^{2}$$

Calculate 12 to the power of 2 and get 144.

$$2x^{2}+16x+64=144$$

Subtract 64 from both sides.

$$2x^{2}+16x=144-64$$

Subtract 64 from 144 to get 80.

$$2x^{2}+16x=80$$

Divide both sides by 2.

$$\frac{2x^{2}+16x}{2}=\frac{80}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{16}{2}x=\frac{80}{2}$$

Divide 16 by 2.

$$x^{2}+8x=\frac{80}{2}$$

Divide 80 by 2.

$$x^{2}+8x=40$$

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.

$$x^{2}+8x+4^{2}=40+4^{2}$$

Square 4.

$$x^{2}+8x+16=40+16$$

Add 40 to 16.

$$x^{2}+8x+16=56$$

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}.

$$\left(x+4\right)^{2}=56$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{56}$$

Simplify.

$$x+4=2\sqrt{14} x+4=-2\sqrt{14}$$

Subtract 4 from both sides of the equation.

$$x=2\sqrt{14}-4 x=-2\sqrt{14}-4$$