Solve z^2+9z-54=(2z+9)(z-6)

$$z ^ { 2 } + 9 z - 54 = ( 2 z + 9 ) ( z - 6 )$$

Solve for z

$$z=12$$

$$z=0$$

Steps Using Factoring

Use the distributive property to multiply 2z+9 by z-6 and combine like terms.

$$z^{2}+9z-54=2z^{2}-3z-54$$

Subtract 2z^{2} from both sides.

$$z^{2}+9z-54-2z^{2}=-3z-54$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+9z-54=-3z-54$$

Add 3z to both sides.

$$-z^{2}+9z-54+3z=-54$$

Combine 9z and 3z to get 12z.

$$-z^{2}+12z-54=-54$$

Add 54 to both sides.

$$-z^{2}+12z-54+54=0$$

Add -54 and 54 to get 0.

$$-z^{2}+12z=0$$

Factor out z.

$$z\left(-z+12\right)=0$$

To find equation solutions, solve z=0 and -z+12=0.

$$z=0 z=12$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2z+9 by z-6 and combine like terms.

$$z^{2}+9z-54=2z^{2}-3z-54$$

Subtract 2z^{2} from both sides.

$$z^{2}+9z-54-2z^{2}=-3z-54$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+9z-54=-3z-54$$

Add 3z to both sides.

$$-z^{2}+9z-54+3z=-54$$

Combine 9z and 3z to get 12z.

$$-z^{2}+12z-54=-54$$

Add 54 to both sides.

$$-z^{2}+12z-54+54=0$$

Add -54 and 54 to get 0.

$$-z^{2}+12z=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 12 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-12±\sqrt{12^{2}}}{2\left(-1\right)}$$

Take the square root of 12^{2}.

$$z=\frac{-12±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{-12±12}{-2}$$

Now solve the equation z=\frac{-12±12}{-2} when ± is plus. Add -12 to 12.

$$z=\frac{0}{-2}$$

Divide 0 by -2.

$$z=0$$

Now solve the equation z=\frac{-12±12}{-2} when ± is minus. Subtract 12 from -12.

$$z=-\frac{24}{-2}$$

Divide -24 by -2.

$$z=12$$

The equation is now solved.

$$z=0 z=12$$

Steps for Completing the Square

Use the distributive property to multiply 2z+9 by z-6 and combine like terms.

$$z^{2}+9z-54=2z^{2}-3z-54$$

Subtract 2z^{2} from both sides.

$$z^{2}+9z-54-2z^{2}=-3z-54$$

Combine z^{2} and -2z^{2} to get -z^{2}.

$$-z^{2}+9z-54=-3z-54$$

Add 3z to both sides.

$$-z^{2}+9z-54+3z=-54$$

Combine 9z and 3z to get 12z.

$$-z^{2}+12z-54=-54$$

Add 54 to both sides.

$$-z^{2}+12z=-54+54$$

Add -54 and 54 to get 0.

$$-z^{2}+12z=0$$

Divide both sides by -1.

$$\frac{-z^{2}+12z}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$z^{2}+\frac{12}{-1}z=\frac{0}{-1}$$

Divide 12 by -1.

$$z^{2}-12z=\frac{0}{-1}$$

Divide 0 by -1.

$$z^{2}-12z=0$$

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.

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

Square -6.

$$z^{2}-12z+36=36$$

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

$$\left(z-6\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-6\right)^{2}}=\sqrt{36}$$

Simplify.

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

Add 6 to both sides of the equation.

$$z=12 z=0$$