Solve sqrt{6z+9}-z=-3

$$\sqrt { 6 z + 9 } - z = - 3$$

Solve for z

$$z=12$$

Solution Steps

Subtract -z from both sides of the equation.

$$\sqrt{6z+9}=-3+z$$

Square both sides of the equation.

$$\left(\sqrt{6z+9}\right)^{2}=\left(-3+z\right)^{2}$$

Calculate \sqrt{6z+9} to the power of 2 and get 6z+9.

$$6z+9=\left(-3+z\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3+z\right)^{2}.

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

Subtract 9 from both sides.

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

Subtract 9 from 9 to get 0.

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

Add 6z to both sides.

$$6z+6z=z^{2}$$

Combine 6z and 6z to get 12z.

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

Subtract z^{2} from both sides.

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

Factor out z.

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

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

$$z=0 z=12$$

Substitute 0 for z in the equation \sqrt{6z+9}-z=-3.

$$\sqrt{6\times 0+9}-0=-3$$

Simplify. The value z=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-3$$

Substitute 12 for z in the equation \sqrt{6z+9}-z=-3.

$$\sqrt{6\times 12+9}-12=-3$$

Simplify. The value z=12 satisfies the equation.

$$-3=-3$$

Equation \sqrt{6z+9}=z-3 has a unique solution.

$$z=12$$