Solve left(-50xright)^2+25x+500=-300

$${ \left(-50x \right) }^{ 2 } +25x+500 = -300$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{12799}i}{200}\approx -0.005+0.565663327i$$

$$x=\frac{-\sqrt{12799}i-1}{200}\approx -0.005-0.565663327i$$

Steps Using the Quadratic Formula

Expand \left(-50x\right)^{2}.

$$\left(-50\right)^{2}x^{2}+25x+500=-300$$

Calculate -50 to the power of 2 and get 2500.

$$2500x^{2}+25x+500=-300$$

Add 300 to both sides.

$$2500x^{2}+25x+500+300=0$$

Add 500 and 300 to get 800.

$$2500x^{2}+25x+800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2500 for a, 25 for b, and 800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\times 2500\times 800}}{2\times 2500}$$

Square 25.

$$x=\frac{-25±\sqrt{625-4\times 2500\times 800}}{2\times 2500}$$

Multiply -4 times 2500.

$$x=\frac{-25±\sqrt{625-10000\times 800}}{2\times 2500}$$

Multiply -10000 times 800.

$$x=\frac{-25±\sqrt{625-8000000}}{2\times 2500}$$

Add 625 to -8000000.

$$x=\frac{-25±\sqrt{-7999375}}{2\times 2500}$$

Take the square root of -7999375.

$$x=\frac{-25±25\sqrt{12799}i}{2\times 2500}$$

Multiply 2 times 2500.

$$x=\frac{-25±25\sqrt{12799}i}{5000}$$

Now solve the equation x=\frac{-25±25\sqrt{12799}i}{5000} when ± is plus. Add -25 to 25i\sqrt{12799}.

$$x=\frac{-25+25\sqrt{12799}i}{5000}$$

Divide -25+25i\sqrt{12799} by 5000.

$$x=\frac{-1+\sqrt{12799}i}{200}$$

Now solve the equation x=\frac{-25±25\sqrt{12799}i}{5000} when ± is minus. Subtract 25i\sqrt{12799} from -25.

$$x=\frac{-25\sqrt{12799}i-25}{5000}$$

Divide -25-25i\sqrt{12799} by 5000.

$$x=\frac{-\sqrt{12799}i-1}{200}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{12799}i}{200} x=\frac{-\sqrt{12799}i-1}{200}$$

Steps for Completing the Square

Expand \left(-50x\right)^{2}.

$$\left(-50\right)^{2}x^{2}+25x+500=-300$$

Calculate -50 to the power of 2 and get 2500.

$$2500x^{2}+25x+500=-300$$

Subtract 500 from both sides.

$$2500x^{2}+25x=-300-500$$

Subtract 500 from -300 to get -800.

$$2500x^{2}+25x=-800$$

Divide both sides by 2500.

$$\frac{2500x^{2}+25x}{2500}=-\frac{800}{2500}$$

Dividing by 2500 undoes the multiplication by 2500.

$$x^{2}+\frac{25}{2500}x=-\frac{800}{2500}$$

Reduce the fraction \frac{25}{2500} to lowest terms by extracting and canceling out 25.

$$x^{2}+\frac{1}{100}x=-\frac{800}{2500}$$

Reduce the fraction \frac{-800}{2500} to lowest terms by extracting and canceling out 100.

$$x^{2}+\frac{1}{100}x=-\frac{8}{25}$$

Divide \frac{1}{100}, the coefficient of the x term, by 2 to get \frac{1}{200}. Then add the square of \frac{1}{200} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{100}x+\left(\frac{1}{200}\right)^{2}=-\frac{8}{25}+\left(\frac{1}{200}\right)^{2}$$

Square \frac{1}{200} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{100}x+\frac{1}{40000}=-\frac{8}{25}+\frac{1}{40000}$$

Add -\frac{8}{25} to \frac{1}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{100}x+\frac{1}{40000}=-\frac{12799}{40000}$$

Factor x^{2}+\frac{1}{100}x+\frac{1}{40000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{200}\right)^{2}=-\frac{12799}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{200}\right)^{2}}=\sqrt{-\frac{12799}{40000}}$$

Simplify.

$$x+\frac{1}{200}=\frac{\sqrt{12799}i}{200} x+\frac{1}{200}=-\frac{\sqrt{12799}i}{200}$$

Subtract \frac{1}{200} from both sides of the equation.

$$x=\frac{-1+\sqrt{12799}i}{200} x=\frac{-\sqrt{12799}i-1}{200}$$