Solve left(-50xright)^2-25x-500=0

$${ \left(-50x \right) }^{ 2 } -25x-500 = 0$$

Solve for x

$$x=\frac{3\sqrt{889}+1}{200}\approx 0.452241545$$

$$x=\frac{1-3\sqrt{889}}{200}\approx -0.442241545$$

Steps Using the Quadratic Formula

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

$$\left(-50\right)^{2}x^{2}-25x-500=0$$

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

$$2500x^{2}-25x-500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2500 for a, -25 for b, and -500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 2500\left(-500\right)}}{2\times 2500}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 2500\left(-500\right)}}{2\times 2500}$$

Multiply -4 times 2500.

$$x=\frac{-\left(-25\right)±\sqrt{625-10000\left(-500\right)}}{2\times 2500}$$

Multiply -10000 times -500.

$$x=\frac{-\left(-25\right)±\sqrt{625+5000000}}{2\times 2500}$$

Add 625 to 5000000.

$$x=\frac{-\left(-25\right)±\sqrt{5000625}}{2\times 2500}$$

Take the square root of 5000625.

$$x=\frac{-\left(-25\right)±75\sqrt{889}}{2\times 2500}$$

The opposite of -25 is 25.

$$x=\frac{25±75\sqrt{889}}{2\times 2500}$$

Multiply 2 times 2500.

$$x=\frac{25±75\sqrt{889}}{5000}$$

Now solve the equation x=\frac{25±75\sqrt{889}}{5000} when ± is plus. Add 25 to 75\sqrt{889}.

$$x=\frac{75\sqrt{889}+25}{5000}$$

Divide 25+75\sqrt{889} by 5000.

$$x=\frac{3\sqrt{889}+1}{200}$$

Now solve the equation x=\frac{25±75\sqrt{889}}{5000} when ± is minus. Subtract 75\sqrt{889} from 25.

$$x=\frac{25-75\sqrt{889}}{5000}$$

Divide 25-75\sqrt{889} by 5000.

$$x=\frac{1-3\sqrt{889}}{200}$$

The equation is now solved.

$$x=\frac{3\sqrt{889}+1}{200} x=\frac{1-3\sqrt{889}}{200}$$

Steps for Completing the Square

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

$$\left(-50\right)^{2}x^{2}-25x-500=0$$

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

$$2500x^{2}-25x-500=0$$

Add 500 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2500.

$$\frac{2500x^{2}-25x}{2500}=\frac{500}{2500}$$

Dividing by 2500 undoes the multiplication by 2500.

$$x^{2}+\left(-\frac{25}{2500}\right)x=\frac{500}{2500}$$

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

$$x^{2}-\frac{1}{100}x=\frac{500}{2500}$$

Reduce the fraction \frac{500}{2500} to lowest terms by extracting and canceling out 500.

$$x^{2}-\frac{1}{100}x=\frac{1}{5}$$

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.

$$x^{2}-\frac{1}{100}x+\left(-\frac{1}{200}\right)^{2}=\frac{1}{5}+\left(-\frac{1}{200}\right)^{2}$$

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

$$x^{2}-\frac{1}{100}x+\frac{1}{40000}=\frac{1}{5}+\frac{1}{40000}$$

Add \frac{1}{5} to \frac{1}{40000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{100}x+\frac{1}{40000}=\frac{8001}{40000}$$

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}.

$$\left(x-\frac{1}{200}\right)^{2}=\frac{8001}{40000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{200}\right)^{2}}=\sqrt{\frac{8001}{40000}}$$

Simplify.

$$x-\frac{1}{200}=\frac{3\sqrt{889}}{200} x-\frac{1}{200}=-\frac{3\sqrt{889}}{200}$$

Add \frac{1}{200} to both sides of the equation.

$$x=\frac{3\sqrt{889}+1}{200} x=\frac{1-3\sqrt{889}}{200}$$