Solve 0=2q^2-4q+1500

$$0 = 2 q ^ { 2 } - 4 q + 1500$$

Solve for q

$$q=1+\sqrt{749}i\approx 1+27.367864367i$$

$$q=-\sqrt{749}i+1\approx 1-27.367864367i$$

Steps Using the Quadratic Formula

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

$$2q^{2}-4q+1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and 1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$q=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 1500}}{2\times 2}$$

Square -4.

$$q=\frac{-\left(-4\right)±\sqrt{16-4\times 2\times 1500}}{2\times 2}$$

Multiply -4 times 2.

$$q=\frac{-\left(-4\right)±\sqrt{16-8\times 1500}}{2\times 2}$$

Multiply -8 times 1500.

$$q=\frac{-\left(-4\right)±\sqrt{16-12000}}{2\times 2}$$

Add 16 to -12000.

$$q=\frac{-\left(-4\right)±\sqrt{-11984}}{2\times 2}$$

Take the square root of -11984.

$$q=\frac{-\left(-4\right)±4\sqrt{749}i}{2\times 2}$$

The opposite of -4 is 4.

$$q=\frac{4±4\sqrt{749}i}{2\times 2}$$

Multiply 2 times 2.

$$q=\frac{4±4\sqrt{749}i}{4}$$

Now solve the equation q=\frac{4±4\sqrt{749}i}{4} when ± is plus. Add 4 to 4i\sqrt{749}.

$$q=\frac{4+4\sqrt{749}i}{4}$$

Divide 4+4i\sqrt{749} by 4.

$$q=1+\sqrt{749}i$$

Now solve the equation q=\frac{4±4\sqrt{749}i}{4} when ± is minus. Subtract 4i\sqrt{749} from 4.

$$q=\frac{-4\sqrt{749}i+4}{4}$$

Divide 4-4i\sqrt{749} by 4.

$$q=-\sqrt{749}i+1$$

The equation is now solved.

$$q=1+\sqrt{749}i q=-\sqrt{749}i+1$$

Steps for Completing the Square

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

$$2q^{2}-4q+1500=0$$

Subtract 1500 from both sides. Anything subtracted from zero gives its negation.

$$2q^{2}-4q=-1500$$

Divide both sides by 2.

$$\frac{2q^{2}-4q}{2}=-\frac{1500}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$q^{2}+\left(-\frac{4}{2}\right)q=-\frac{1500}{2}$$

Divide -4 by 2.

$$q^{2}-2q=-\frac{1500}{2}$$

Divide -1500 by 2.

$$q^{2}-2q=-750$$

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.

$$q^{2}-2q+1=-750+1$$

Add -750 to 1.

$$q^{2}-2q+1=-749$$

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}.

$$\left(q-1\right)^{2}=-749$$

Take the square root of both sides of the equation.

$$\sqrt{\left(q-1\right)^{2}}=\sqrt{-749}$$

Simplify.

$$q-1=\sqrt{749}i q-1=-\sqrt{749}i$$

Add 1 to both sides of the equation.

$$q=1+\sqrt{749}i q=-\sqrt{749}i+1$$