Solve 0.50x=20x^2+240x+800

$$0.50 x = 20 x ^ { 2 } + 240 x + 800$$

Solve for x (complex solution)

$$x=\frac{-479+i\times 3\sqrt{2951}}{80}\approx -5.9875+2.037116528i$$

$$x=\frac{-i\times 3\sqrt{2951}-479}{80}\approx -5.9875-2.037116528i$$

Steps Using the Quadratic Formula

Subtract 20x^{2} from both sides.

$$0.5x-20x^{2}=240x+800$$

Subtract 240x from both sides.

$$0.5x-20x^{2}-240x=800$$

Combine 0.5x and -240x to get -239.5x.

$$-239.5x-20x^{2}=800$$

Subtract 800 from both sides.

$$-239.5x-20x^{2}-800=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-20x^{2}-239.5x-800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, -239.5 for b, and -800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-239.5\right)±\sqrt{\left(-239.5\right)^{2}-4\left(-20\right)\left(-800\right)}}{2\left(-20\right)}$$

Square -239.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-239.5\right)±\sqrt{57360.25-4\left(-20\right)\left(-800\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-\left(-239.5\right)±\sqrt{57360.25+80\left(-800\right)}}{2\left(-20\right)}$$

Multiply 80 times -800.

$$x=\frac{-\left(-239.5\right)±\sqrt{57360.25-64000}}{2\left(-20\right)}$$

Add 57360.25 to -64000.

$$x=\frac{-\left(-239.5\right)±\sqrt{-6639.75}}{2\left(-20\right)}$$

Take the square root of -6639.75.

$$x=\frac{-\left(-239.5\right)±\frac{3\sqrt{2951}i}{2}}{2\left(-20\right)}$$

The opposite of -239.5 is 239.5.

$$x=\frac{239.5±\frac{3\sqrt{2951}i}{2}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{239.5±\frac{3\sqrt{2951}i}{2}}{-40}$$

Now solve the equation x=\frac{239.5±\frac{3\sqrt{2951}i}{2}}{-40} when ± is plus. Add 239.5 to \frac{3i\sqrt{2951}}{2}.

$$x=\frac{479+3\sqrt{2951}i}{-40\times 2}$$

Divide \frac{479+3i\sqrt{2951}}{2} by -40.

$$x=\frac{-3\sqrt{2951}i-479}{80}$$

Now solve the equation x=\frac{239.5±\frac{3\sqrt{2951}i}{2}}{-40} when ± is minus. Subtract \frac{3i\sqrt{2951}}{2} from 239.5.

$$x=\frac{-3\sqrt{2951}i+479}{-40\times 2}$$

Divide \frac{479-3i\sqrt{2951}}{2} by -40.

$$x=\frac{-479+3\sqrt{2951}i}{80}$$

The equation is now solved.

$$x=\frac{-3\sqrt{2951}i-479}{80} x=\frac{-479+3\sqrt{2951}i}{80}$$

Steps for Completing the Square

Subtract 20x^{2} from both sides.

$$0.5x-20x^{2}=240x+800$$

Subtract 240x from both sides.

$$0.5x-20x^{2}-240x=800$$

Combine 0.5x and -240x to get -239.5x.

$$-239.5x-20x^{2}=800$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-20x^{2}-239.5x=800$$

Divide both sides by -20.

$$\frac{-20x^{2}-239.5x}{-20}=\frac{800}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\left(-\frac{239.5}{-20}\right)x=\frac{800}{-20}$$

Divide -239.5 by -20.

$$x^{2}+11.975x=\frac{800}{-20}$$

Divide 800 by -20.

$$x^{2}+11.975x=-40$$

Divide 11.975, the coefficient of the x term, by 2 to get 5.9875. Then add the square of 5.9875 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+11.975x+5.9875^{2}=-40+5.9875^{2}$$

Square 5.9875 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+11.975x+35.85015625=-40+35.85015625$$

Add -40 to 35.85015625.

$$x^{2}+11.975x+35.85015625=-4.14984375$$

Factor x^{2}+11.975x+35.85015625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5.9875\right)^{2}=-4.14984375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5.9875\right)^{2}}=\sqrt{-4.14984375}$$

Simplify.

$$x+5.9875=\frac{3\sqrt{2951}i}{80} x+5.9875=-\frac{3\sqrt{2951}i}{80}$$

Subtract 5.9875 from both sides of the equation.

$$x=\frac{-479+3\sqrt{2951}i}{80} x=\frac{-3\sqrt{2951}i-479}{80}$$