Solve z^2-8z=0quad1-2^2

$$z ^ { 2 } - 8 z = 0 \quad 1 - 2 ^ { 2 }$$

Solve for z

$$z=2\sqrt{3}+4\approx 7.464101615$$

$$z=4-2\sqrt{3}\approx 0.535898385$$

Steps Using the Quadratic Formula

Multiply 0 and 1 to get 0.

$$z^{2}-8z=0-2^{2}$$

Calculate 2 to the power of 2 and get 4.

$$z^{2}-8z=0-4$$

Subtract 4 from 0 to get -4.

$$z^{2}-8z=-4$$

Add 4 to both sides.

$$z^{2}-8z+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 4}}{2}$$

Square -8.

$$z=\frac{-\left(-8\right)±\sqrt{64-4\times 4}}{2}$$

Multiply -4 times 4.

$$z=\frac{-\left(-8\right)±\sqrt{64-16}}{2}$$

Add 64 to -16.

$$z=\frac{-\left(-8\right)±\sqrt{48}}{2}$$

Take the square root of 48.

$$z=\frac{-\left(-8\right)±4\sqrt{3}}{2}$$

The opposite of -8 is 8.

$$z=\frac{8±4\sqrt{3}}{2}$$

Now solve the equation z=\frac{8±4\sqrt{3}}{2} when ± is plus. Add 8 to 4\sqrt{3}.

$$z=\frac{4\sqrt{3}+8}{2}$$

Divide 8+4\sqrt{3} by 2.

$$z=2\sqrt{3}+4$$

Now solve the equation z=\frac{8±4\sqrt{3}}{2} when ± is minus. Subtract 4\sqrt{3} from 8.

$$z=\frac{8-4\sqrt{3}}{2}$$

Divide 8-4\sqrt{3} by 2.

$$z=4-2\sqrt{3}$$

The equation is now solved.

$$z=2\sqrt{3}+4 z=4-2\sqrt{3}$$

Steps for Completing the Square

Multiply 0 and 1 to get 0.

$$z^{2}-8z=0-2^{2}$$

Calculate 2 to the power of 2 and get 4.

$$z^{2}-8z=0-4$$

Subtract 4 from 0 to get -4.

$$z^{2}-8z=-4$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-8z+\left(-4\right)^{2}=-4+\left(-4\right)^{2}$$

Square -4.

$$z^{2}-8z+16=-4+16$$

Add -4 to 16.

$$z^{2}-8z+16=12$$

Factor z^{2}-8z+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-4\right)^{2}=12$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-4\right)^{2}}=\sqrt{12}$$

Simplify.

$$z-4=2\sqrt{3} z-4=-2\sqrt{3}$$

Add 4 to both sides of the equation.

$$z=2\sqrt{3}+4 z=4-2\sqrt{3}$$