Solve x^2+(12-x)^2=348

$$x ^ { 2 } + ( 12 - x ) ^ { 2 } = 348$$

Solve for x

$$x=\sqrt{138}+6\approx 17.747340124$$

$$x=6-\sqrt{138}\approx -5.747340124$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$x^{2}+144-24x+x^{2}=348$$

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

$$2x^{2}+144-24x=348$$

Subtract 348 from both sides.

$$2x^{2}+144-24x-348=0$$

Subtract 348 from 144 to get -204.

$$2x^{2}-204-24x=0$$

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.

$$2x^{2}-24x-204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and -204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\left(-204\right)}}{2\times 2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\left(-204\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\left(-204\right)}}{2\times 2}$$

Multiply -8 times -204.

$$x=\frac{-\left(-24\right)±\sqrt{576+1632}}{2\times 2}$$

Add 576 to 1632.

$$x=\frac{-\left(-24\right)±\sqrt{2208}}{2\times 2}$$

Take the square root of 2208.

$$x=\frac{-\left(-24\right)±4\sqrt{138}}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{138}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±4\sqrt{138}}{4}$$

Now solve the equation x=\frac{24±4\sqrt{138}}{4} when ± is plus. Add 24 to 4\sqrt{138}.

$$x=\frac{4\sqrt{138}+24}{4}$$

Divide 24+4\sqrt{138} by 4.

$$x=\sqrt{138}+6$$

Now solve the equation x=\frac{24±4\sqrt{138}}{4} when ± is minus. Subtract 4\sqrt{138} from 24.

$$x=\frac{24-4\sqrt{138}}{4}$$

Divide 24-4\sqrt{138} by 4.

$$x=6-\sqrt{138}$$

The equation is now solved.

$$x=\sqrt{138}+6 x=6-\sqrt{138}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$x^{2}+144-24x+x^{2}=348$$

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

$$2x^{2}+144-24x=348$$

Subtract 144 from both sides.

$$2x^{2}-24x=348-144$$

Subtract 144 from 348 to get 204.

$$2x^{2}-24x=204$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=\frac{204}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=\frac{204}{2}$$

Divide -24 by 2.

$$x^{2}-12x=\frac{204}{2}$$

Divide 204 by 2.

$$x^{2}-12x=102$$

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.

$$x^{2}-12x+\left(-6\right)^{2}=102+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=102+36$$

Add 102 to 36.

$$x^{2}-12x+36=138$$

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}.

$$\left(x-6\right)^{2}=138$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{138}$$

Simplify.

$$x-6=\sqrt{138} x-6=-\sqrt{138}$$

Add 6 to both sides of the equation.

$$x=\sqrt{138}+6 x=6-\sqrt{138}$$