Solve x^2-406x+163^2=0

$${ x }^{ 2 } -406x+ { 163 }^{ 2 } =0$$

Solve for x

$$x=4\sqrt{915}+203\approx 323.995867698$$

$$x=203-4\sqrt{915}\approx 82.004132302$$

Steps Using the Quadratic Formula

Calculate 163 to the power of 2 and get 26569.

$$x^{2}-406x+26569=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -406 for b, and 26569 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-406\right)±\sqrt{\left(-406\right)^{2}-4\times 26569}}{2}$$

Square -406.

$$x=\frac{-\left(-406\right)±\sqrt{164836-4\times 26569}}{2}$$

Multiply -4 times 26569.

$$x=\frac{-\left(-406\right)±\sqrt{164836-106276}}{2}$$

Add 164836 to -106276.

$$x=\frac{-\left(-406\right)±\sqrt{58560}}{2}$$

Take the square root of 58560.

$$x=\frac{-\left(-406\right)±8\sqrt{915}}{2}$$

The opposite of -406 is 406.

$$x=\frac{406±8\sqrt{915}}{2}$$

Now solve the equation x=\frac{406±8\sqrt{915}}{2} when ± is plus. Add 406 to 8\sqrt{915}.

$$x=\frac{8\sqrt{915}+406}{2}$$

Divide 406+8\sqrt{915} by 2.

$$x=4\sqrt{915}+203$$

Now solve the equation x=\frac{406±8\sqrt{915}}{2} when ± is minus. Subtract 8\sqrt{915} from 406.

$$x=\frac{406-8\sqrt{915}}{2}$$

Divide 406-8\sqrt{915} by 2.

$$x=203-4\sqrt{915}$$

The equation is now solved.

$$x=4\sqrt{915}+203 x=203-4\sqrt{915}$$

Steps for Completing the Square

Calculate 163 to the power of 2 and get 26569.

$$x^{2}-406x+26569=0$$

Subtract 26569 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-406x=-26569$$

Divide -406, the coefficient of the x term, by 2 to get -203. Then add the square of -203 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-406x+\left(-203\right)^{2}=-26569+\left(-203\right)^{2}$$

Square -203.

$$x^{2}-406x+41209=-26569+41209$$

Add -26569 to 41209.

$$x^{2}-406x+41209=14640$$

Factor x^{2}-406x+41209. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-203\right)^{2}=14640$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-203\right)^{2}}=\sqrt{14640}$$

Simplify.

$$x-203=4\sqrt{915} x-203=-4\sqrt{915}$$

Add 203 to both sides of the equation.

$$x=4\sqrt{915}+203 x=203-4\sqrt{915}$$