Solve -2/3x^2-400x-25800=0

$$- \frac{ 2 }{ 3 } { x }^{ 2 } -400x-25800=0$$

Solve for x

$$x=30\sqrt{57}-300\approx -73.504966942$$

$$x=-30\sqrt{57}-300\approx -526.495033058$$

Steps Using the Quadratic Formula

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.

$$-\frac{2}{3}x^{2}-400x-25800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{3} for a, -400 for b, and -25800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-400\right)±\sqrt{\left(-400\right)^{2}-4\left(-\frac{2}{3}\right)\left(-25800\right)}}{2\left(-\frac{2}{3}\right)}$$

Square -400.

$$x=\frac{-\left(-400\right)±\sqrt{160000-4\left(-\frac{2}{3}\right)\left(-25800\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply -4 times -\frac{2}{3}.

$$x=\frac{-\left(-400\right)±\sqrt{160000+\frac{8}{3}\left(-25800\right)}}{2\left(-\frac{2}{3}\right)}$$

Multiply \frac{8}{3} times -25800.

$$x=\frac{-\left(-400\right)±\sqrt{160000-68800}}{2\left(-\frac{2}{3}\right)}$$

Add 160000 to -68800.

$$x=\frac{-\left(-400\right)±\sqrt{91200}}{2\left(-\frac{2}{3}\right)}$$

Take the square root of 91200.

$$x=\frac{-\left(-400\right)±40\sqrt{57}}{2\left(-\frac{2}{3}\right)}$$

The opposite of -400 is 400.

$$x=\frac{400±40\sqrt{57}}{2\left(-\frac{2}{3}\right)}$$

Multiply 2 times -\frac{2}{3}.

$$x=\frac{400±40\sqrt{57}}{-\frac{4}{3}}$$

Now solve the equation x=\frac{400±40\sqrt{57}}{-\frac{4}{3}} when ± is plus. Add 400 to 40\sqrt{57}.

$$x=\frac{40\sqrt{57}+400}{-\frac{4}{3}}$$

Divide 400+40\sqrt{57} by -\frac{4}{3} by multiplying 400+40\sqrt{57} by the reciprocal of -\frac{4}{3}.

$$x=-30\sqrt{57}-300$$

Now solve the equation x=\frac{400±40\sqrt{57}}{-\frac{4}{3}} when ± is minus. Subtract 40\sqrt{57} from 400.

$$x=\frac{400-40\sqrt{57}}{-\frac{4}{3}}$$

Divide 400-40\sqrt{57} by -\frac{4}{3} by multiplying 400-40\sqrt{57} by the reciprocal of -\frac{4}{3}.

$$x=30\sqrt{57}-300$$

The equation is now solved.

$$x=-30\sqrt{57}-300 x=30\sqrt{57}-300$$

Steps for Completing the Square

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.

$$-\frac{2}{3}x^{2}-400x-25800=0$$

Add 25800 to both sides of the equation.

$$-\frac{2}{3}x^{2}-400x-25800-\left(-25800\right)=-\left(-25800\right)$$

Subtracting -25800 from itself leaves 0.

$$-\frac{2}{3}x^{2}-400x=-\left(-25800\right)$$

Subtract -25800 from 0.

$$-\frac{2}{3}x^{2}-400x=25800$$

Divide both sides of the equation by -\frac{2}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{3}x^{2}-400x}{-\frac{2}{3}}=\frac{25800}{-\frac{2}{3}}$$

Dividing by -\frac{2}{3} undoes the multiplication by -\frac{2}{3}.

$$x^{2}+\left(-\frac{400}{-\frac{2}{3}}\right)x=\frac{25800}{-\frac{2}{3}}$$

Divide -400 by -\frac{2}{3} by multiplying -400 by the reciprocal of -\frac{2}{3}.

$$x^{2}+600x=\frac{25800}{-\frac{2}{3}}$$

Divide 25800 by -\frac{2}{3} by multiplying 25800 by the reciprocal of -\frac{2}{3}.

$$x^{2}+600x=-38700$$

Divide 600, the coefficient of the x term, by 2 to get 300. Then add the square of 300 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+600x+300^{2}=-38700+300^{2}$$

Square 300.

$$x^{2}+600x+90000=-38700+90000$$

Add -38700 to 90000.

$$x^{2}+600x+90000=51300$$

Factor x^{2}+600x+90000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+300\right)^{2}=51300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+300\right)^{2}}=\sqrt{51300}$$

Simplify.

$$x+300=30\sqrt{57} x+300=-30\sqrt{57}$$

Subtract 300 from both sides of the equation.

$$x=30\sqrt{57}-300 x=-30\sqrt{57}-300$$