Solve x^2+left(12-xright)^2=378

$${ x }^{ 2 } + { \left(12-x \right) }^{ 2 } = 378$$

Solve for x

$$x=3\sqrt{17}+6\approx 18.369316877$$

$$x=6-3\sqrt{17}\approx -6.369316877$$

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}.

$$x^{2}+144-24x+x^{2}=378$$

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

$$2x^{2}+144-24x=378$$

Subtract 378 from both sides.

$$2x^{2}+144-24x-378=0$$

Subtract 378 from 144 to get -234.

$$2x^{2}-234-24x=0$$

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.

$$2x^{2}-24x-234=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -24 for b, and -234 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\left(-234\right)}}{2\times 2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\left(-234\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\left(-234\right)}}{2\times 2}$$

Multiply -8 times -234.

$$x=\frac{-\left(-24\right)±\sqrt{576+1872}}{2\times 2}$$

Add 576 to 1872.

$$x=\frac{-\left(-24\right)±\sqrt{2448}}{2\times 2}$$

Take the square root of 2448.

$$x=\frac{-\left(-24\right)±12\sqrt{17}}{2\times 2}$$

The opposite of -24 is 24.

$$x=\frac{24±12\sqrt{17}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{24±12\sqrt{17}}{4}$$

Now solve the equation x=\frac{24±12\sqrt{17}}{4} when ± is plus. Add 24 to 12\sqrt{17}.

$$x=\frac{12\sqrt{17}+24}{4}$$

Divide 24+12\sqrt{17} by 4.

$$x=3\sqrt{17}+6$$

Now solve the equation x=\frac{24±12\sqrt{17}}{4} when ± is minus. Subtract 12\sqrt{17} from 24.

$$x=\frac{24-12\sqrt{17}}{4}$$

Divide 24-12\sqrt{17} by 4.

$$x=6-3\sqrt{17}$$

The equation is now solved.

$$x=3\sqrt{17}+6 x=6-3\sqrt{17}$$

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}.

$$x^{2}+144-24x+x^{2}=378$$

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

$$2x^{2}+144-24x=378$$

Subtract 144 from both sides.

$$2x^{2}-24x=378-144$$

Subtract 144 from 378 to get 234.

$$2x^{2}-24x=234$$

Divide both sides by 2.

$$\frac{2x^{2}-24x}{2}=\frac{234}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=\frac{234}{2}$$

Divide -24 by 2.

$$x^{2}-12x=\frac{234}{2}$$

Divide 234 by 2.

$$x^{2}-12x=117$$

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.

$$x^{2}-12x+\left(-6\right)^{2}=117+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=117+36$$

Add 117 to 36.

$$x^{2}-12x+36=153$$

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}.

$$\left(x-6\right)^{2}=153$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{153}$$

Simplify.

$$x-6=3\sqrt{17} x-6=-3\sqrt{17}$$

Add 6 to both sides of the equation.

$$x=3\sqrt{17}+6 x=6-3\sqrt{17}$$