Solve x^2+(68-x)^2=157

$$x ^ { 2 } + ( 68 - x ) ^ { 2 } = 157$$

Solve for x (complex solution)

$$x=\frac{\sqrt{4310}i}{2}+34\approx 34+32.825295124i$$

$$x=-\frac{\sqrt{4310}i}{2}+34\approx 34-32.825295124i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(68-x\right)^{2}.

$$x^{2}+4624-136x+x^{2}=157$$

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

$$2x^{2}+4624-136x=157$$

Subtract 157 from both sides.

$$2x^{2}+4624-136x-157=0$$

Subtract 157 from 4624 to get 4467.

$$2x^{2}+4467-136x=0$$

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.

$$2x^{2}-136x+4467=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -136 for b, and 4467 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-136\right)±\sqrt{\left(-136\right)^{2}-4\times 2\times 4467}}{2\times 2}$$

Square -136.

$$x=\frac{-\left(-136\right)±\sqrt{18496-4\times 2\times 4467}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-136\right)±\sqrt{18496-8\times 4467}}{2\times 2}$$

Multiply -8 times 4467.

$$x=\frac{-\left(-136\right)±\sqrt{18496-35736}}{2\times 2}$$

Add 18496 to -35736.

$$x=\frac{-\left(-136\right)±\sqrt{-17240}}{2\times 2}$$

Take the square root of -17240.

$$x=\frac{-\left(-136\right)±2\sqrt{4310}i}{2\times 2}$$

The opposite of -136 is 136.

$$x=\frac{136±2\sqrt{4310}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{136±2\sqrt{4310}i}{4}$$

Now solve the equation x=\frac{136±2\sqrt{4310}i}{4} when ± is plus. Add 136 to 2i\sqrt{4310}.

$$x=\frac{136+2\sqrt{4310}i}{4}$$

Divide 136+2i\sqrt{4310} by 4.

$$x=\frac{\sqrt{4310}i}{2}+34$$

Now solve the equation x=\frac{136±2\sqrt{4310}i}{4} when ± is minus. Subtract 2i\sqrt{4310} from 136.

$$x=\frac{-2\sqrt{4310}i+136}{4}$$

Divide 136-2i\sqrt{4310} by 4.

$$x=-\frac{\sqrt{4310}i}{2}+34$$

The equation is now solved.

$$x=\frac{\sqrt{4310}i}{2}+34 x=-\frac{\sqrt{4310}i}{2}+34$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(68-x\right)^{2}.

$$x^{2}+4624-136x+x^{2}=157$$

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

$$2x^{2}+4624-136x=157$$

Subtract 4624 from both sides.

$$2x^{2}-136x=157-4624$$

Subtract 4624 from 157 to get -4467.

$$2x^{2}-136x=-4467$$

Divide both sides by 2.

$$\frac{2x^{2}-136x}{2}=-\frac{4467}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{136}{2}\right)x=-\frac{4467}{2}$$

Divide -136 by 2.

$$x^{2}-68x=-\frac{4467}{2}$$

Divide -68, the coefficient of the x term, by 2 to get -34. Then add the square of -34 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-68x+\left(-34\right)^{2}=-\frac{4467}{2}+\left(-34\right)^{2}$$

Square -34.

$$x^{2}-68x+1156=-\frac{4467}{2}+1156$$

Add -\frac{4467}{2} to 1156.

$$x^{2}-68x+1156=-\frac{2155}{2}$$

Factor x^{2}-68x+1156. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-34\right)^{2}=-\frac{2155}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{-\frac{2155}{2}}$$

Simplify.

$$x-34=\frac{\sqrt{4310}i}{2} x-34=-\frac{\sqrt{4310}i}{2}$$

Add 34 to both sides of the equation.

$$x=\frac{\sqrt{4310}i}{2}+34 x=-\frac{\sqrt{4310}i}{2}+34$$