Solve (2k-8)^2-16k=0

$$( 2 k - 8 ) ^ { 2 } - 16 k = 0$$

Solve for k

$$k=2\sqrt{5}+6\approx 10.472135955$$

$$k=6-2\sqrt{5}\approx 1.527864045$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2k-8\right)^{2}.

$$4k^{2}-32k+64-16k=0$$

Combine -32k and -16k to get -48k.

$$4k^{2}-48k+64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -48 for b, and 64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 4\times 64}}{2\times 4}$$

Square -48.

$$k=\frac{-\left(-48\right)±\sqrt{2304-4\times 4\times 64}}{2\times 4}$$

Multiply -4 times 4.

$$k=\frac{-\left(-48\right)±\sqrt{2304-16\times 64}}{2\times 4}$$

Multiply -16 times 64.

$$k=\frac{-\left(-48\right)±\sqrt{2304-1024}}{2\times 4}$$

Add 2304 to -1024.

$$k=\frac{-\left(-48\right)±\sqrt{1280}}{2\times 4}$$

Take the square root of 1280.

$$k=\frac{-\left(-48\right)±16\sqrt{5}}{2\times 4}$$

The opposite of -48 is 48.

$$k=\frac{48±16\sqrt{5}}{2\times 4}$$

Multiply 2 times 4.

$$k=\frac{48±16\sqrt{5}}{8}$$

Now solve the equation k=\frac{48±16\sqrt{5}}{8} when ± is plus. Add 48 to 16\sqrt{5}.

$$k=\frac{16\sqrt{5}+48}{8}$$

Divide 48+16\sqrt{5} by 8.

$$k=2\sqrt{5}+6$$

Now solve the equation k=\frac{48±16\sqrt{5}}{8} when ± is minus. Subtract 16\sqrt{5} from 48.

$$k=\frac{48-16\sqrt{5}}{8}$$

Divide 48-16\sqrt{5} by 8.

$$k=6-2\sqrt{5}$$

The equation is now solved.

$$k=2\sqrt{5}+6 k=6-2\sqrt{5}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2k-8\right)^{2}.

$$4k^{2}-32k+64-16k=0$$

Combine -32k and -16k to get -48k.

$$4k^{2}-48k+64=0$$

Subtract 64 from both sides. Anything subtracted from zero gives its negation.

$$4k^{2}-48k=-64$$

Divide both sides by 4.

$$\frac{4k^{2}-48k}{4}=-\frac{64}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$k^{2}+\left(-\frac{48}{4}\right)k=-\frac{64}{4}$$

Divide -48 by 4.

$$k^{2}-12k=-\frac{64}{4}$$

Divide -64 by 4.

$$k^{2}-12k=-16$$

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.

$$k^{2}-12k+\left(-6\right)^{2}=-16+\left(-6\right)^{2}$$

Square -6.

$$k^{2}-12k+36=-16+36$$

Add -16 to 36.

$$k^{2}-12k+36=20$$

Factor k^{2}-12k+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}.

$$\left(k-6\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-6\right)^{2}}=\sqrt{20}$$

Simplify.

$$k-6=2\sqrt{5} k-6=-2\sqrt{5}$$

Add 6 to both sides of the equation.

$$k=2\sqrt{5}+6 k=6-2\sqrt{5}$$