Solve 3z(3z+10)=24

$$3 z ( 3 z + 10 ) = 24$$

Solve for z

$$z=-4$$

$$z=\frac{2}{3}\approx 0.666666667$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3z by 3z+10.

$$9z^{2}+30z=24$$

Subtract 24 from both sides.

$$9z^{2}+30z-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 30 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-30±\sqrt{30^{2}-4\times 9\left(-24\right)}}{2\times 9}$$

Square 30.

$$z=\frac{-30±\sqrt{900-4\times 9\left(-24\right)}}{2\times 9}$$

Multiply -4 times 9.

$$z=\frac{-30±\sqrt{900-36\left(-24\right)}}{2\times 9}$$

Multiply -36 times -24.

$$z=\frac{-30±\sqrt{900+864}}{2\times 9}$$

Add 900 to 864.

$$z=\frac{-30±\sqrt{1764}}{2\times 9}$$

Take the square root of 1764.

$$z=\frac{-30±42}{2\times 9}$$

Multiply 2 times 9.

$$z=\frac{-30±42}{18}$$

Now solve the equation z=\frac{-30±42}{18} when ± is plus. Add -30 to 42.

$$z=\frac{12}{18}$$

Reduce the fraction \frac{12}{18} to lowest terms by extracting and canceling out 6.

$$z=\frac{2}{3}$$

Now solve the equation z=\frac{-30±42}{18} when ± is minus. Subtract 42 from -30.

$$z=-\frac{72}{18}$$

Divide -72 by 18.

$$z=-4$$

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 3z by 3z+10.

$$9z^{2}+30z=24$$

Divide both sides by 9.

$$\frac{9z^{2}+30z}{9}=\frac{24}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$z^{2}+\frac{30}{9}z=\frac{24}{9}$$

Reduce the fraction \frac{30}{9} to lowest terms by extracting and canceling out 3.

$$z^{2}+\frac{10}{3}z=\frac{24}{9}$$

Reduce the fraction \frac{24}{9} to lowest terms by extracting and canceling out 3.

$$z^{2}+\frac{10}{3}z=\frac{8}{3}$$

Divide \frac{10}{3}, the coefficient of the x term, by 2 to get \frac{5}{3}. Then add the square of \frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+\frac{10}{3}z+\left(\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(\frac{5}{3}\right)^{2}$$

Square \frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$z^{2}+\frac{10}{3}z+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Add \frac{8}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}+\frac{10}{3}z+\frac{25}{9}=\frac{49}{9}$$

Factor z^{2}+\frac{10}{3}z+\frac{25}{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}.

$$\left(z+\frac{5}{3}\right)^{2}=\frac{49}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Simplify.

$$z+\frac{5}{3}=\frac{7}{3} z+\frac{5}{3}=-\frac{7}{3}$$

Subtract \frac{5}{3} from both sides of the equation.

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