Solve 3z(z-9)+9(z-9)=0

$$3 z ( z - 9 ) + 9 ( z - 9 ) = 0$$

Solve for z

$$z=-3$$

$$z=9$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 3z by z-9.

$$3z^{2}-27z+9\left(z-9\right)=0$$

Use the distributive property to multiply 9 by z-9.

$$3z^{2}-27z+9z-81=0$$

Combine -27z and 9z to get -18z.

$$3z^{2}-18z-81=0$$

Divide both sides by 3.

$$z^{2}-6z-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as z^{2}+az+bz-27. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-27\right)=-27$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -27.

$$1,-27 3,-9$$

Calculate the sum for each pair.

$$1-27=-26 3-9=-6$$

The solution is the pair that gives sum -6.

$$a=-9 b=3$$

Rewrite z^{2}-6z-27 as \left(z^{2}-9z\right)+\left(3z-27\right).

$$\left(z^{2}-9z\right)+\left(3z-27\right)$$

Factor out z in the first and 3 in the second group.

$$z\left(z-9\right)+3\left(z-9\right)$$

Factor out common term z-9 by using distributive property.

$$\left(z-9\right)\left(z+3\right)$$

To find equation solutions, solve z-9=0 and z+3=0.

$$z=9 z=-3$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3z by z-9.

$$3z^{2}-27z+9\left(z-9\right)=0$$

Use the distributive property to multiply 9 by z-9.

$$3z^{2}-27z+9z-81=0$$

Combine -27z and 9z to get -18z.

$$3z^{2}-18z-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -18 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\left(-81\right)}}{2\times 3}$$

Square -18.

$$z=\frac{-\left(-18\right)±\sqrt{324-4\times 3\left(-81\right)}}{2\times 3}$$

Multiply -4 times 3.

$$z=\frac{-\left(-18\right)±\sqrt{324-12\left(-81\right)}}{2\times 3}$$

Multiply -12 times -81.

$$z=\frac{-\left(-18\right)±\sqrt{324+972}}{2\times 3}$$

Add 324 to 972.

$$z=\frac{-\left(-18\right)±\sqrt{1296}}{2\times 3}$$

Take the square root of 1296.

$$z=\frac{-\left(-18\right)±36}{2\times 3}$$

The opposite of -18 is 18.

$$z=\frac{18±36}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{18±36}{6}$$

Now solve the equation z=\frac{18±36}{6} when ± is plus. Add 18 to 36.

$$z=\frac{54}{6}$$

Divide 54 by 6.

$$z=9$$

Now solve the equation z=\frac{18±36}{6} when ± is minus. Subtract 36 from 18.

$$z=-\frac{18}{6}$$

Divide -18 by 6.

$$z=-3$$

The equation is now solved.

$$z=9 z=-3$$

Steps for Completing the Square

Use the distributive property to multiply 3z by z-9.

$$3z^{2}-27z+9\left(z-9\right)=0$$

Use the distributive property to multiply 9 by z-9.

$$3z^{2}-27z+9z-81=0$$

Combine -27z and 9z to get -18z.

$$3z^{2}-18z-81=0$$

Add 81 to both sides. Anything plus zero gives itself.

$$3z^{2}-18z=81$$

Divide both sides by 3.

$$\frac{3z^{2}-18z}{3}=\frac{81}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$z^{2}+\left(-\frac{18}{3}\right)z=\frac{81}{3}$$

Divide -18 by 3.

$$z^{2}-6z=\frac{81}{3}$$

Divide 81 by 3.

$$z^{2}-6z=27$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-6z+\left(-3\right)^{2}=27+\left(-3\right)^{2}$$

Square -3.

$$z^{2}-6z+9=27+9$$

Add 27 to 9.

$$z^{2}-6z+9=36$$

Factor z^{2}-6z+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}.

$$\left(z-3\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-3\right)^{2}}=\sqrt{36}$$

Simplify.

$$z-3=6 z-3=-6$$

Add 3 to both sides of the equation.

$$z=9 z=-3$$