Solve z^2+3z-30=(2z+5)(z+6)

$$z ^ { 2 } + 3 z - 30 = ( 2 z + 5 ) ( z + 6 )$$

Solve for z

$$z=-\sqrt{11}i-7\approx -7-3.31662479i$$

$$z=-7+\sqrt{11}i\approx -7+3.31662479i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2z+5 by z+6 and combine like terms.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Subtract 2z^{2} from both sides.

$$z^{2}+3z-30-2z^{2}=17z+30$$

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

$$-z^{2}+3z-30=17z+30$$

Subtract 17z from both sides.

$$-z^{2}+3z-30-17z=30$$

Combine 3z and -17z to get -14z.

$$-z^{2}-14z-30=30$$

Subtract 30 from both sides.

$$-z^{2}-14z-30-30=0$$

Subtract 30 from -30 to get -60.

$$-z^{2}-14z-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -14 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Square -14.

$$z=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\left(-60\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$z=\frac{-\left(-14\right)±\sqrt{196+4\left(-60\right)}}{2\left(-1\right)}$$

Multiply 4 times -60.

$$z=\frac{-\left(-14\right)±\sqrt{196-240}}{2\left(-1\right)}$$

Add 196 to -240.

$$z=\frac{-\left(-14\right)±\sqrt{-44}}{2\left(-1\right)}$$

Take the square root of -44.

$$z=\frac{-\left(-14\right)±2\sqrt{11}i}{2\left(-1\right)}$$

The opposite of -14 is 14.

$$z=\frac{14±2\sqrt{11}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$z=\frac{14±2\sqrt{11}i}{-2}$$

Now solve the equation z=\frac{14±2\sqrt{11}i}{-2} when ± is plus. Add 14 to 2i\sqrt{11}.

$$z=\frac{14+2\sqrt{11}i}{-2}$$

Divide 14+2i\sqrt{11} by -2.

$$z=-\sqrt{11}i-7$$

Now solve the equation z=\frac{14±2\sqrt{11}i}{-2} when ± is minus. Subtract 2i\sqrt{11} from 14.

$$z=\frac{-2\sqrt{11}i+14}{-2}$$

Divide 14-2i\sqrt{11} by -2.

$$z=-7+\sqrt{11}i$$

The equation is now solved.

$$z=-\sqrt{11}i-7 z=-7+\sqrt{11}i$$

Steps for Completing the Square

Use the distributive property to multiply 2z+5 by z+6 and combine like terms.

$$z^{2}+3z-30=2z^{2}+17z+30$$

Subtract 2z^{2} from both sides.

$$z^{2}+3z-30-2z^{2}=17z+30$$

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

$$-z^{2}+3z-30=17z+30$$

Subtract 17z from both sides.

$$-z^{2}+3z-30-17z=30$$

Combine 3z and -17z to get -14z.

$$-z^{2}-14z-30=30$$

Add 30 to both sides.

$$-z^{2}-14z=30+30$$

Add 30 and 30 to get 60.

$$-z^{2}-14z=60$$

Divide both sides by -1.

$$\frac{-z^{2}-14z}{-1}=\frac{60}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$z^{2}+\left(-\frac{14}{-1}\right)z=\frac{60}{-1}$$

Divide -14 by -1.

$$z^{2}+14z=\frac{60}{-1}$$

Divide 60 by -1.

$$z^{2}+14z=-60$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+14z+7^{2}=-60+7^{2}$$

Square 7.

$$z^{2}+14z+49=-60+49$$

Add -60 to 49.

$$z^{2}+14z+49=-11$$

Factor z^{2}+14z+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+7\right)^{2}=-11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+7\right)^{2}}=\sqrt{-11}$$

Simplify.

$$z+7=\sqrt{11}i z+7=-\sqrt{11}i$$

Subtract 7 from both sides of the equation.

$$z=-7+\sqrt{11}i z=-\sqrt{11}i-7$$