Solve 2x(2x-10)+2x=2000

$$2x(2x-10)+2x=2000$$

Solve for x

$$x = \frac{\sqrt{8081} + 9}{4} \approx 24.723595618$$

$$x=\frac{9-\sqrt{8081}}{4}\approx -20.223595618$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 2x-10.

$$4x^{2}-20x+2x=2000$$

Combine -20x and 2x to get -18x.

$$4x^{2}-18x=2000$$

Subtract 2000 from both sides.

$$4x^{2}-18x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -18 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 4\left(-2000\right)}}{2\times 4}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 4\left(-2000\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-18\right)±\sqrt{324-16\left(-2000\right)}}{2\times 4}$$

Multiply -16 times -2000.

$$x=\frac{-\left(-18\right)±\sqrt{324+32000}}{2\times 4}$$

Add 324 to 32000.

$$x=\frac{-\left(-18\right)±\sqrt{32324}}{2\times 4}$$

Take the square root of 32324.

$$x=\frac{-\left(-18\right)±2\sqrt{8081}}{2\times 4}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{8081}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{18±2\sqrt{8081}}{8}$$

Now solve the equation x=\frac{18±2\sqrt{8081}}{8} when ± is plus. Add 18 to 2\sqrt{8081}.

$$x=\frac{2\sqrt{8081}+18}{8}$$

Divide 18+2\sqrt{8081} by 8.

$$x=\frac{\sqrt{8081}+9}{4}$$

Now solve the equation x=\frac{18±2\sqrt{8081}}{8} when ± is minus. Subtract 2\sqrt{8081} from 18.

$$x=\frac{18-2\sqrt{8081}}{8}$$

Divide 18-2\sqrt{8081} by 8.

$$x=\frac{9-\sqrt{8081}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{8081}+9}{4} x=\frac{9-\sqrt{8081}}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 2x-10.

$$4x^{2}-20x+2x=2000$$

Combine -20x and 2x to get -18x.

$$4x^{2}-18x=2000$$

Divide both sides by 4.

$$\frac{4x^{2}-18x}{4}=\frac{2000}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{18}{4}\right)x=\frac{2000}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{9}{2}x=\frac{2000}{4}$$

Divide 2000 by 4.

$$x^{2}-\frac{9}{2}x=500$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=500+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=500+\frac{81}{16}$$

Add 500 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{8081}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{8081}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{8081}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{\sqrt{8081}}{4} x-\frac{9}{4}=-\frac{\sqrt{8081}}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{\sqrt{8081}+9}{4} x=\frac{9-\sqrt{8081}}{4}$$