Solve -96/x/1-x=40.5

$$\frac{ \frac{ -96 }{ x } }{ 1-x } =40.5$$

Solve for x

$$x=\frac{\sqrt{849}}{18}+0.5\approx 2.118755809$$

$$x=-\frac{\sqrt{849}}{18}+0.5\approx -1.118755809$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

$$\frac{-96}{x}=40.5\left(-x+1\right)$$

Use the distributive property to multiply 40.5 by -x+1.

$$\frac{-96}{x}=-40.5x+40.5$$

Add 40.5x to both sides.

$$\frac{-96}{x}+40.5x=40.5$$

Subtract 40.5 from both sides.

$$\frac{-96}{x}+40.5x-40.5=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-96+40.5xx+x\left(-40.5\right)=0$$

Multiply x and x to get x^{2}.

$$-96+40.5x^{2}+x\left(-40.5\right)=0$$

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.

$$40.5x^{2}-40.5x-96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40.5 for a, -40.5 for b, and -96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40.5\right)±\sqrt{\left(-40.5\right)^{2}-4\times 40.5\left(-96\right)}}{2\times 40.5}$$

Square -40.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-40.5\right)±\sqrt{1640.25-4\times 40.5\left(-96\right)}}{2\times 40.5}$$

Multiply -4 times 40.5.

$$x=\frac{-\left(-40.5\right)±\sqrt{1640.25-162\left(-96\right)}}{2\times 40.5}$$

Multiply -162 times -96.

$$x=\frac{-\left(-40.5\right)±\sqrt{1640.25+15552}}{2\times 40.5}$$

Add 1640.25 to 15552.

$$x=\frac{-\left(-40.5\right)±\sqrt{17192.25}}{2\times 40.5}$$

Take the square root of 17192.25.

$$x=\frac{-\left(-40.5\right)±\frac{9\sqrt{849}}{2}}{2\times 40.5}$$

The opposite of -40.5 is 40.5.

$$x=\frac{40.5±\frac{9\sqrt{849}}{2}}{2\times 40.5}$$

Multiply 2 times 40.5.

$$x=\frac{40.5±\frac{9\sqrt{849}}{2}}{81}$$

Now solve the equation x=\frac{40.5±\frac{9\sqrt{849}}{2}}{81} when ± is plus. Add 40.5 to \frac{9\sqrt{849}}{2}.

$$x=\frac{9\sqrt{849}+81}{2\times 81}$$

Divide \frac{81+9\sqrt{849}}{2} by 81.

$$x=\frac{\sqrt{849}}{18}+\frac{1}{2}$$

Now solve the equation x=\frac{40.5±\frac{9\sqrt{849}}{2}}{81} when ± is minus. Subtract \frac{9\sqrt{849}}{2} from 40.5.

$$x=\frac{81-9\sqrt{849}}{2\times 81}$$

Divide \frac{81-9\sqrt{849}}{2} by 81.

$$x=-\frac{\sqrt{849}}{18}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{849}}{18}+\frac{1}{2} x=-\frac{\sqrt{849}}{18}+\frac{1}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -x+1.

$$\frac{-96}{x}=40.5\left(-x+1\right)$$

Use the distributive property to multiply 40.5 by -x+1.

$$\frac{-96}{x}=-40.5x+40.5$$

Add 40.5x to both sides.

$$\frac{-96}{x}+40.5x=40.5$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-96+40.5xx=40.5x$$

Multiply x and x to get x^{2}.

$$-96+40.5x^{2}=40.5x$$

Subtract 40.5x from both sides.

$$-96+40.5x^{2}-40.5x=0$$

Add 96 to both sides. Anything plus zero gives itself.

$$40.5x^{2}-40.5x=96$$

Divide both sides of the equation by 40.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{40.5x^{2}-40.5x}{40.5}=\frac{96}{40.5}$$

Dividing by 40.5 undoes the multiplication by 40.5.

$$x^{2}+\left(-\frac{40.5}{40.5}\right)x=\frac{96}{40.5}$$

Divide -40.5 by 40.5 by multiplying -40.5 by the reciprocal of 40.5.

$$x^{2}-x=\frac{96}{40.5}$$

Divide 96 by 40.5 by multiplying 96 by the reciprocal of 40.5.

$$x^{2}-x=\frac{64}{27}$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{64}{27}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=\frac{64}{27}+\frac{1}{4}$$

Add \frac{64}{27} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{283}{108}$$

Factor x^{2}-x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{283}{108}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{283}{108}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{849}}{18} x-\frac{1}{2}=-\frac{\sqrt{849}}{18}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{849}}{18}+\frac{1}{2} x=-\frac{\sqrt{849}}{18}+\frac{1}{2}$$