Solve (x-4)^2+(-x+8)^2=80

$$( x - 4 ) ^ { 2 } + ( - x + 8 ) ^ { 2 } = 80$$

Solve for x

$$x=12$$

$$x=0$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16+\left(-x+8\right)^{2}=80$$

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

$$x^{2}-8x+16+\left(-x\right)^{2}+16\left(-x\right)+64=80$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-8x+16+x^{2}+16\left(-x\right)+64=80$$

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

$$2x^{2}-8x+16+16\left(-x\right)+64=80$$

Add 16 and 64 to get 80.

$$2x^{2}-8x+80+16\left(-x\right)=80$$

Subtract 80 from both sides.

$$2x^{2}-8x+80+16\left(-x\right)-80=0$$

Subtract 80 from 80 to get 0.

$$2x^{2}-8x+16\left(-x\right)=0$$

Multiply 16 and -1 to get -16.

$$2x^{2}-8x-16x=0$$

Combine -8x and -16x to get -24x.

$$2x^{2}-24x=0$$

Factor out x.

$$x\left(2x-24\right)=0$$

To find equation solutions, solve x=0 and 2x-24=0.

$$x=0 x=12$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16+\left(-x+8\right)^{2}=80$$

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

$$x^{2}-8x+16+\left(-x\right)^{2}+16\left(-x\right)+64=80$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-8x+16+x^{2}+16\left(-x\right)+64=80$$

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

$$2x^{2}-8x+16+16\left(-x\right)+64=80$$

Add 16 and 64 to get 80.

$$2x^{2}-8x+80+16\left(-x\right)=80$$

Subtract 80 from both sides.

$$2x^{2}-8x+80+16\left(-x\right)-80=0$$

Subtract 80 from 80 to get 0.

$$2x^{2}-8x+16\left(-x\right)=0$$

Multiply 16 and -1 to get -16.

$$2x^{2}-8x-16x=0$$

Combine -8x and -16x to get -24x.

$$2x^{2}-24x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}}}{2\times 2}$$

Take the square root of \left(-24\right)^{2}.

$$x=\frac{-\left(-24\right)±24}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±24}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±24}{4}$$

Now solve the equation x=\frac{24±24}{4} when ± is plus. Add 24 to 24.

$$x=\frac{48}{4}$$

Divide 48 by 4.

$$x=12$$

Now solve the equation x=\frac{24±24}{4} when ± is minus. Subtract 24 from 24.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

The equation is now solved.

$$x=12 x=0$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$x^{2}-8x+16+\left(-x+8\right)^{2}=80$$

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

$$x^{2}-8x+16+\left(-x\right)^{2}+16\left(-x\right)+64=80$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}-8x+16+x^{2}+16\left(-x\right)+64=80$$

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

$$2x^{2}-8x+16+16\left(-x\right)+64=80$$

Add 16 and 64 to get 80.

$$2x^{2}-8x+80+16\left(-x\right)=80$$

Subtract 80 from both sides.

$$2x^{2}-8x+16\left(-x\right)=80-80$$

Subtract 80 from 80 to get 0.

$$2x^{2}-8x+16\left(-x\right)=0$$

Multiply 16 and -1 to get -16.

$$2x^{2}-8x-16x=0$$

Combine -8x and -16x to get -24x.

$$2x^{2}-24x=0$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=\frac{0}{2}$$

Divide -24 by 2.

$$x^{2}-12x=\frac{0}{2}$$

Divide 0 by 2.

$$x^{2}-12x=0$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=36$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-6=6 x-6=-6$$

Add 6 to both sides of the equation.

$$x=12 x=0$$