Solve (x-1)^2=16x

$$( x - 1 ) ^ { 2 } = 16 x$$

Solve for x

$$x=4\sqrt{5}+9\approx 17.94427191$$

$$x=9-4\sqrt{5}\approx 0.05572809$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=16x$$

Subtract 16x from both sides.

$$x^{2}-2x+1-16x=0$$

Combine -2x and -16x to get -18x.

$$x^{2}-18x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4}}{2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4}}{2}$$

Add 324 to -4.

$$x=\frac{-\left(-18\right)±\sqrt{320}}{2}$$

Take the square root of 320.

$$x=\frac{-\left(-18\right)±8\sqrt{5}}{2}$$

The opposite of -18 is 18.

$$x=\frac{18±8\sqrt{5}}{2}$$

Now solve the equation x=\frac{18±8\sqrt{5}}{2} when ± is plus. Add 18 to 8\sqrt{5}.

$$x=\frac{8\sqrt{5}+18}{2}$$

Divide 18+8\sqrt{5} by 2.

$$x=4\sqrt{5}+9$$

Now solve the equation x=\frac{18±8\sqrt{5}}{2} when ± is minus. Subtract 8\sqrt{5} from 18.

$$x=\frac{18-8\sqrt{5}}{2}$$

Divide 18-8\sqrt{5} by 2.

$$x=9-4\sqrt{5}$$

The equation is now solved.

$$x=4\sqrt{5}+9 x=9-4\sqrt{5}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

$$x^{2}-2x+1=16x$$

Subtract 16x from both sides.

$$x^{2}-2x+1-16x=0$$

Combine -2x and -16x to get -18x.

$$x^{2}-18x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-18x=-1$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-1+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-1+81$$

Add -1 to 81.

$$x^{2}-18x+81=80$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=80$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{80}$$

Simplify.

$$x-9=4\sqrt{5} x-9=-4\sqrt{5}$$

Add 9 to both sides of the equation.

$$x=4\sqrt{5}+9 x=9-4\sqrt{5}$$