Solve x^2+(8-x)^2=36

$$x ^ { 2 } + ( 8 - x ) ^ { 2 } = 36$$

Solve for x

$$x=\sqrt{2}+4\approx 5.414213562$$

$$x=4-\sqrt{2}\approx 2.585786438$$

Steps Using the Quadratic Formula

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

$$x^{2}+64-16x+x^{2}=36$$

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

$$2x^{2}+64-16x=36$$

Subtract 36 from both sides.

$$2x^{2}+64-16x-36=0$$

Subtract 36 from 64 to get 28.

$$2x^{2}+28-16x=0$$

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.

$$2x^{2}-16x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -16 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 2\times 28}}{2\times 2}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 2\times 28}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-16\right)±\sqrt{256-8\times 28}}{2\times 2}$$

Multiply -8 times 28.

$$x=\frac{-\left(-16\right)±\sqrt{256-224}}{2\times 2}$$

Add 256 to -224.

$$x=\frac{-\left(-16\right)±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

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

The opposite of -16 is 16.

$$x=\frac{16±4\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{16±4\sqrt{2}}{4}$$

Now solve the equation x=\frac{16±4\sqrt{2}}{4} when ± is plus. Add 16 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}+16}{4}$$

Divide 16+4\sqrt{2} by 4.

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

Now solve the equation x=\frac{16±4\sqrt{2}}{4} when ± is minus. Subtract 4\sqrt{2} from 16.

$$x=\frac{16-4\sqrt{2}}{4}$$

Divide 16-4\sqrt{2} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$x^{2}+64-16x+x^{2}=36$$

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

$$2x^{2}+64-16x=36$$

Subtract 64 from both sides.

$$2x^{2}-16x=36-64$$

Subtract 64 from 36 to get -28.

$$2x^{2}-16x=-28$$

Divide both sides by 2.

$$\frac{2x^{2}-16x}{2}=-\frac{28}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{16}{2}\right)x=-\frac{28}{2}$$

Divide -16 by 2.

$$x^{2}-8x=-\frac{28}{2}$$

Divide -28 by 2.

$$x^{2}-8x=-14$$

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.

$$x^{2}-8x+\left(-4\right)^{2}=-14+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-14+16$$

Add -14 to 16.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 4 to both sides of the equation.

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