Solve 899x^2-250/3x+23725/9=0

$$899 { x }^{ 2 } - \frac{ 250 }{ 3 } x+ \frac{ 23725 }{ 9 } =0$$

Solve for x (complex solution)

$$x=\frac{125+5\sqrt{852526}i}{2697}\approx 0.046347794+1.711760002i$$

$$x=\frac{-5\sqrt{852526}i+125}{2697}\approx 0.046347794-1.711760002i$$

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.

$$899x^{2}-\frac{250}{3}x+\frac{23725}{9}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 899 for a, -\frac{250}{3} for b, and \frac{23725}{9} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{250}{3}\right)±\sqrt{\left(-\frac{250}{3}\right)^{2}-4\times 899\times \frac{23725}{9}}}{2\times 899}$$

Square -\frac{250}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{250}{3}\right)±\sqrt{\frac{62500}{9}-4\times 899\times \frac{23725}{9}}}{2\times 899}$$

Multiply -4 times 899.

$$x=\frac{-\left(-\frac{250}{3}\right)±\sqrt{\frac{62500}{9}-3596\times \frac{23725}{9}}}{2\times 899}$$

Multiply -3596 times \frac{23725}{9}.

$$x=\frac{-\left(-\frac{250}{3}\right)±\sqrt{\frac{62500-85315100}{9}}}{2\times 899}$$

Add \frac{62500}{9} to -\frac{85315100}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{250}{3}\right)±\sqrt{-\frac{85252600}{9}}}{2\times 899}$$

Take the square root of -\frac{85252600}{9}.

$$x=\frac{-\left(-\frac{250}{3}\right)±\frac{10\sqrt{852526}i}{3}}{2\times 899}$$

The opposite of -\frac{250}{3} is \frac{250}{3}.

$$x=\frac{\frac{250}{3}±\frac{10\sqrt{852526}i}{3}}{2\times 899}$$

Multiply 2 times 899.

$$x=\frac{\frac{250}{3}±\frac{10\sqrt{852526}i}{3}}{1798}$$

Now solve the equation x=\frac{\frac{250}{3}±\frac{10\sqrt{852526}i}{3}}{1798} when ± is plus. Add \frac{250}{3} to \frac{10i\sqrt{852526}}{3}.

$$x=\frac{250+10\sqrt{852526}i}{3\times 1798}$$

Divide \frac{250+10i\sqrt{852526}}{3} by 1798.

$$x=\frac{125+5\sqrt{852526}i}{2697}$$

Now solve the equation x=\frac{\frac{250}{3}±\frac{10\sqrt{852526}i}{3}}{1798} when ± is minus. Subtract \frac{10i\sqrt{852526}}{3} from \frac{250}{3}.

$$x=\frac{-10\sqrt{852526}i+250}{3\times 1798}$$

Divide \frac{250-10i\sqrt{852526}}{3} by 1798.

$$x=\frac{-5\sqrt{852526}i+125}{2697}$$

The equation is now solved.

$$x=\frac{125+5\sqrt{852526}i}{2697} x=\frac{-5\sqrt{852526}i+125}{2697}$$

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.

$$899x^{2}-\frac{250}{3}x+\frac{23725}{9}=0$$

Subtract \frac{23725}{9} from both sides of the equation.

$$899x^{2}-\frac{250}{3}x+\frac{23725}{9}-\frac{23725}{9}=-\frac{23725}{9}$$

Subtracting \frac{23725}{9} from itself leaves 0.

$$899x^{2}-\frac{250}{3}x=-\frac{23725}{9}$$

Divide both sides by 899.

$$\frac{899x^{2}-\frac{250}{3}x}{899}=-\frac{\frac{23725}{9}}{899}$$

Dividing by 899 undoes the multiplication by 899.

$$x^{2}+\left(-\frac{\frac{250}{3}}{899}\right)x=-\frac{\frac{23725}{9}}{899}$$

Divide -\frac{250}{3} by 899.

$$x^{2}-\frac{250}{2697}x=-\frac{\frac{23725}{9}}{899}$$

Divide -\frac{23725}{9} by 899.

$$x^{2}-\frac{250}{2697}x=-\frac{23725}{8091}$$

Divide -\frac{250}{2697}, the coefficient of the x term, by 2 to get -\frac{125}{2697}. Then add the square of -\frac{125}{2697} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{250}{2697}x+\left(-\frac{125}{2697}\right)^{2}=-\frac{23725}{8091}+\left(-\frac{125}{2697}\right)^{2}$$

Square -\frac{125}{2697} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{250}{2697}x+\frac{15625}{7273809}=-\frac{23725}{8091}+\frac{15625}{7273809}$$

Add -\frac{23725}{8091} to \frac{15625}{7273809} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{250}{2697}x+\frac{15625}{7273809}=-\frac{21313150}{7273809}$$

Factor x^{2}-\frac{250}{2697}x+\frac{15625}{7273809}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{125}{2697}\right)^{2}=-\frac{21313150}{7273809}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{125}{2697}\right)^{2}}=\sqrt{-\frac{21313150}{7273809}}$$

Simplify.

$$x-\frac{125}{2697}=\frac{5\sqrt{852526}i}{2697} x-\frac{125}{2697}=-\frac{5\sqrt{852526}i}{2697}$$

Add \frac{125}{2697} to both sides of the equation.

$$x=\frac{125+5\sqrt{852526}i}{2697} x=\frac{-5\sqrt{852526}i+125}{2697}$$