Solve -2x^2-20x+250/3x^2+30x=0

$$\frac { - 2 x ^ { 2 } - 20 x + 250 } { 3 x ^ { 2 } + 30 x } = 0$$

Solve for x

$$x=5\sqrt{6}-5\approx 7.247448714$$

$$x=-5\sqrt{6}-5\approx -17.247448714$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+10\right).

$$-2x^{2}-20x+250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -20 for b, and 250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\left(-2\right)\times 250}}{2\left(-2\right)}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\left(-2\right)\times 250}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-20\right)±\sqrt{400+8\times 250}}{2\left(-2\right)}$$

Multiply 8 times 250.

$$x=\frac{-\left(-20\right)±\sqrt{400+2000}}{2\left(-2\right)}$$

Add 400 to 2000.

$$x=\frac{-\left(-20\right)±\sqrt{2400}}{2\left(-2\right)}$$

Take the square root of 2400.

$$x=\frac{-\left(-20\right)±20\sqrt{6}}{2\left(-2\right)}$$

The opposite of -20 is 20.

$$x=\frac{20±20\sqrt{6}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{20±20\sqrt{6}}{-4}$$

Now solve the equation x=\frac{20±20\sqrt{6}}{-4} when ± is plus. Add 20 to 20\sqrt{6}.

$$x=\frac{20\sqrt{6}+20}{-4}$$

Divide 20+20\sqrt{6} by -4.

$$x=-5\sqrt{6}-5$$

Now solve the equation x=\frac{20±20\sqrt{6}}{-4} when ± is minus. Subtract 20\sqrt{6} from 20.

$$x=\frac{20-20\sqrt{6}}{-4}$$

Divide 20-20\sqrt{6} by -4.

$$x=5\sqrt{6}-5$$

The equation is now solved.

$$x=-5\sqrt{6}-5 x=5\sqrt{6}-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+10\right).

$$-2x^{2}-20x+250=0$$

Subtract 250 from both sides. Anything subtracted from zero gives its negation.

$$-2x^{2}-20x=-250$$

Divide both sides by -2.

$$\frac{-2x^{2}-20x}{-2}=-\frac{250}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{20}{-2}\right)x=-\frac{250}{-2}$$

Divide -20 by -2.

$$x^{2}+10x=-\frac{250}{-2}$$

Divide -250 by -2.

$$x^{2}+10x=125$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=125+5^{2}$$

Square 5.

$$x^{2}+10x+25=125+25$$

Add 125 to 25.

$$x^{2}+10x+25=150$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=150$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{150}$$

Simplify.

$$x+5=5\sqrt{6} x+5=-5\sqrt{6}$$

Subtract 5 from both sides of the equation.

$$x=5\sqrt{6}-5 x=-5\sqrt{6}-5$$