Solve 18000=-q^2-2q+534

$$18000 = - q ^ { 2 } - 2 q + 534$$

Solve for q

$$q=-\sqrt{17465}i-1\approx -1-132.155211778i$$

$$q=-1+\sqrt{17465}i\approx -1+132.155211778i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-q^{2}-2q+534=18000$$

Subtract 18000 from both sides.

$$-q^{2}-2q+534-18000=0$$

Subtract 18000 from 534 to get -17466.

$$-q^{2}-2q-17466=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and -17466 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$q=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\left(-17466\right)}}{2\left(-1\right)}$$

Square -2.

$$q=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\left(-17466\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$q=\frac{-\left(-2\right)±\sqrt{4+4\left(-17466\right)}}{2\left(-1\right)}$$

Multiply 4 times -17466.

$$q=\frac{-\left(-2\right)±\sqrt{4-69864}}{2\left(-1\right)}$$

Add 4 to -69864.

$$q=\frac{-\left(-2\right)±\sqrt{-69860}}{2\left(-1\right)}$$

Take the square root of -69860.

$$q=\frac{-\left(-2\right)±2\sqrt{17465}i}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$q=\frac{2±2\sqrt{17465}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$q=\frac{2±2\sqrt{17465}i}{-2}$$

Now solve the equation q=\frac{2±2\sqrt{17465}i}{-2} when ± is plus. Add 2 to 2i\sqrt{17465}.

$$q=\frac{2+2\sqrt{17465}i}{-2}$$

Divide 2+2i\sqrt{17465} by -2.

$$q=-\sqrt{17465}i-1$$

Now solve the equation q=\frac{2±2\sqrt{17465}i}{-2} when ± is minus. Subtract 2i\sqrt{17465} from 2.

$$q=\frac{-2\sqrt{17465}i+2}{-2}$$

Divide 2-2i\sqrt{17465} by -2.

$$q=-1+\sqrt{17465}i$$

The equation is now solved.

$$q=-\sqrt{17465}i-1 q=-1+\sqrt{17465}i$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-q^{2}-2q+534=18000$$

Subtract 534 from both sides.

$$-q^{2}-2q=18000-534$$

Subtract 534 from 18000 to get 17466.

$$-q^{2}-2q=17466$$

Divide both sides by -1.

$$\frac{-q^{2}-2q}{-1}=\frac{17466}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$q^{2}+\left(-\frac{2}{-1}\right)q=\frac{17466}{-1}$$

Divide -2 by -1.

$$q^{2}+2q=\frac{17466}{-1}$$

Divide 17466 by -1.

$$q^{2}+2q=-17466$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$q^{2}+2q+1^{2}=-17466+1^{2}$$

Square 1.

$$q^{2}+2q+1=-17466+1$$

Add -17466 to 1.

$$q^{2}+2q+1=-17465$$

Factor q^{2}+2q+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(q+1\right)^{2}=-17465$$

Take the square root of both sides of the equation.

$$\sqrt{\left(q+1\right)^{2}}=\sqrt{-17465}$$

Simplify.

$$q+1=\sqrt{17465}i q+1=-\sqrt{17465}i$$

Subtract 1 from both sides of the equation.

$$q=-1+\sqrt{17465}i q=-\sqrt{17465}i-1$$