Solve x^2/308-x=83176*10^-5

$$\frac{ { x }^{ 2 } }{ 308-x } = 83176 \times { 10 }^{ -5 }$$

Solve for x

$$x = \frac{\sqrt{160221897609} - 10397}{25000} \approx 15.595211036$$

$$x=\frac{-\sqrt{160221897609}-10397}{25000}\approx -16.426971036$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 308 since division by zero is not defined. Multiply both sides of the equation by -x+308.

$$x^{2}=83176\times 10^{-5}\left(-x+308\right)$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$x^{2}=83176\times \frac{1}{100000}\left(-x+308\right)$$

Multiply 83176 and \frac{1}{100000} to get \frac{10397}{12500}.

$$x^{2}=\frac{10397}{12500}\left(-x+308\right)$$

Use the distributive property to multiply \frac{10397}{12500} by -x+308.

$$x^{2}=-\frac{10397}{12500}x+\frac{800569}{3125}$$

Add \frac{10397}{12500}x to both sides.

$$x^{2}+\frac{10397}{12500}x=\frac{800569}{3125}$$

Subtract \frac{800569}{3125} from both sides.

$$x^{2}+\frac{10397}{12500}x-\frac{800569}{3125}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, \frac{10397}{12500} for b, and -\frac{800569}{3125} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{10397}{12500}±\sqrt{\left(\frac{10397}{12500}\right)^{2}-4\left(-\frac{800569}{3125}\right)}}{2}$$

Square \frac{10397}{12500} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{10397}{12500}±\sqrt{\frac{108097609}{156250000}-4\left(-\frac{800569}{3125}\right)}}{2}$$

Multiply -4 times -\frac{800569}{3125}.

$$x=\frac{-\frac{10397}{12500}±\sqrt{\frac{108097609}{156250000}+\frac{3202276}{3125}}}{2}$$

Add \frac{108097609}{156250000} to \frac{3202276}{3125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{10397}{12500}±\sqrt{\frac{160221897609}{156250000}}}{2}$$

Take the square root of \frac{160221897609}{156250000}.

$$x=\frac{-\frac{10397}{12500}±\frac{\sqrt{160221897609}}{12500}}{2}$$

Now solve the equation x=\frac{-\frac{10397}{12500}±\frac{\sqrt{160221897609}}{12500}}{2} when ± is plus. Add -\frac{10397}{12500} to \frac{\sqrt{160221897609}}{12500}.

$$x=\frac{\sqrt{160221897609}-10397}{2\times 12500}$$

Divide \frac{-10397+\sqrt{160221897609}}{12500} by 2.

$$x=\frac{\sqrt{160221897609}-10397}{25000}$$

Now solve the equation x=\frac{-\frac{10397}{12500}±\frac{\sqrt{160221897609}}{12500}}{2} when ± is minus. Subtract \frac{\sqrt{160221897609}}{12500} from -\frac{10397}{12500}.

$$x=\frac{-\sqrt{160221897609}-10397}{2\times 12500}$$

Divide \frac{-10397-\sqrt{160221897609}}{12500} by 2.

$$x=\frac{-\sqrt{160221897609}-10397}{25000}$$

The equation is now solved.

$$x=\frac{\sqrt{160221897609}-10397}{25000} x=\frac{-\sqrt{160221897609}-10397}{25000}$$

Steps for Completing the Square

Variable x cannot be equal to 308 since division by zero is not defined. Multiply both sides of the equation by -x+308.

$$x^{2}=83176\times 10^{-5}\left(-x+308\right)$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$x^{2}=83176\times \frac{1}{100000}\left(-x+308\right)$$

Multiply 83176 and \frac{1}{100000} to get \frac{10397}{12500}.

$$x^{2}=\frac{10397}{12500}\left(-x+308\right)$$

Use the distributive property to multiply \frac{10397}{12500} by -x+308.

$$x^{2}=-\frac{10397}{12500}x+\frac{800569}{3125}$$

Add \frac{10397}{12500}x to both sides.

$$x^{2}+\frac{10397}{12500}x=\frac{800569}{3125}$$

Divide \frac{10397}{12500}, the coefficient of the x term, by 2 to get \frac{10397}{25000}. Then add the square of \frac{10397}{25000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{10397}{12500}x+\left(\frac{10397}{25000}\right)^{2}=\frac{800569}{3125}+\left(\frac{10397}{25000}\right)^{2}$$

Square \frac{10397}{25000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{10397}{12500}x+\frac{108097609}{625000000}=\frac{800569}{3125}+\frac{108097609}{625000000}$$

Add \frac{800569}{3125} to \frac{108097609}{625000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{10397}{12500}x+\frac{108097609}{625000000}=\frac{160221897609}{625000000}$$

Factor x^{2}+\frac{10397}{12500}x+\frac{108097609}{625000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{10397}{25000}\right)^{2}=\frac{160221897609}{625000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{10397}{25000}\right)^{2}}=\sqrt{\frac{160221897609}{625000000}}$$

Simplify.

$$x+\frac{10397}{25000}=\frac{\sqrt{160221897609}}{25000} x+\frac{10397}{25000}=-\frac{\sqrt{160221897609}}{25000}$$

Subtract \frac{10397}{25000} from both sides of the equation.

$$x=\frac{\sqrt{160221897609}-10397}{25000} x=\frac{-\sqrt{160221897609}-10397}{25000}$$