Solve -4(2z-6)=-3(z-4)z

$$- 4 ( 2 z - 6 ) = - 3 ( z - 4 ) z$$

Solve for z

$$z = \frac{2 \sqrt{7} + 10}{3} \approx 5.097167541$$

$$z = \frac{10 - 2 \sqrt{7}}{3} \approx 1.569499126$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -4 by 2z-6.

$$-8z+24=-3\left(z-4\right)z$$

Use the distributive property to multiply -3 by z-4.

$$-8z+24=\left(-3z+12\right)z$$

Use the distributive property to multiply -3z+12 by z.

$$-8z+24=-3z^{2}+12z$$

Add 3z^{2} to both sides.

$$-8z+24+3z^{2}=12z$$

Subtract 12z from both sides.

$$-8z+24+3z^{2}-12z=0$$

Combine -8z and -12z to get -20z.

$$-20z+24+3z^{2}=0$$

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.

$$3z^{2}-20z+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -20 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 3\times 24}}{2\times 3}$$

Square -20.

$$z=\frac{-\left(-20\right)±\sqrt{400-4\times 3\times 24}}{2\times 3}$$

Multiply -4 times 3.

$$z=\frac{-\left(-20\right)±\sqrt{400-12\times 24}}{2\times 3}$$

Multiply -12 times 24.

$$z=\frac{-\left(-20\right)±\sqrt{400-288}}{2\times 3}$$

Add 400 to -288.

$$z=\frac{-\left(-20\right)±\sqrt{112}}{2\times 3}$$

Take the square root of 112.

$$z=\frac{-\left(-20\right)±4\sqrt{7}}{2\times 3}$$

The opposite of -20 is 20.

$$z=\frac{20±4\sqrt{7}}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{20±4\sqrt{7}}{6}$$

Now solve the equation z=\frac{20±4\sqrt{7}}{6} when ± is plus. Add 20 to 4\sqrt{7}.

$$z=\frac{4\sqrt{7}+20}{6}$$

Divide 20+4\sqrt{7} by 6.

$$z=\frac{2\sqrt{7}+10}{3}$$

Now solve the equation z=\frac{20±4\sqrt{7}}{6} when ± is minus. Subtract 4\sqrt{7} from 20.

$$z=\frac{20-4\sqrt{7}}{6}$$

Divide 20-4\sqrt{7} by 6.

$$z=\frac{10-2\sqrt{7}}{3}$$

The equation is now solved.

$$z=\frac{2\sqrt{7}+10}{3} z=\frac{10-2\sqrt{7}}{3}$$

Steps for Completing the Square

Use the distributive property to multiply -4 by 2z-6.

$$-8z+24=-3\left(z-4\right)z$$

Use the distributive property to multiply -3 by z-4.

$$-8z+24=\left(-3z+12\right)z$$

Use the distributive property to multiply -3z+12 by z.

$$-8z+24=-3z^{2}+12z$$

Add 3z^{2} to both sides.

$$-8z+24+3z^{2}=12z$$

Subtract 12z from both sides.

$$-8z+24+3z^{2}-12z=0$$

Combine -8z and -12z to get -20z.

$$-20z+24+3z^{2}=0$$

Subtract 24 from both sides. Anything subtracted from zero gives its negation.

$$-20z+3z^{2}=-24$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$3z^{2}-20z=-24$$

Divide both sides by 3.

$$\frac{3z^{2}-20z}{3}=-\frac{24}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$z^{2}-\frac{20}{3}z=-\frac{24}{3}$$

Divide -24 by 3.

$$z^{2}-\frac{20}{3}z=-8$$

Divide -\frac{20}{3}, the coefficient of the x term, by 2 to get -\frac{10}{3}. Then add the square of -\frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{20}{3}z+\left(-\frac{10}{3}\right)^{2}=-8+\left(-\frac{10}{3}\right)^{2}$$

Square -\frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{20}{3}z+\frac{100}{9}=-8+\frac{100}{9}$$

Add -8 to \frac{100}{9}.

$$z^{2}-\frac{20}{3}z+\frac{100}{9}=\frac{28}{9}$$

Factor z^{2}-\frac{20}{3}z+\frac{100}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{10}{3}\right)^{2}=\frac{28}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{10}{3}\right)^{2}}=\sqrt{\frac{28}{9}}$$

Simplify.

$$z-\frac{10}{3}=\frac{2\sqrt{7}}{3} z-\frac{10}{3}=-\frac{2\sqrt{7}}{3}$$

Add \frac{10}{3} to both sides of the equation.

$$z=\frac{2\sqrt{7}+10}{3} z=\frac{10-2\sqrt{7}}{3}$$