Solve 240x=360/x-3

$$240 x = \frac { 360 } { x } - 3$$

Solve for x

$$x = \frac{\sqrt{38401} - 1}{160} \approx 1.218510818$$

$$x=\frac{-\sqrt{38401}-1}{160}\approx -1.231010818$$

Steps Using the Quadratic Formula

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

$$240xx=360+x\left(-3\right)$$

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

$$240x^{2}=360+x\left(-3\right)$$

Subtract 360 from both sides.

$$240x^{2}-360=x\left(-3\right)$$

Subtract x\left(-3\right) from both sides.

$$240x^{2}-360-x\left(-3\right)=0$$

Multiply -1 and -3 to get 3.

$$240x^{2}-360+3x=0$$

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.

$$240x^{2}+3x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 240 for a, 3 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 240\left(-360\right)}}{2\times 240}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 240\left(-360\right)}}{2\times 240}$$

Multiply -4 times 240.

$$x=\frac{-3±\sqrt{9-960\left(-360\right)}}{2\times 240}$$

Multiply -960 times -360.

$$x=\frac{-3±\sqrt{9+345600}}{2\times 240}$$

Add 9 to 345600.

$$x=\frac{-3±\sqrt{345609}}{2\times 240}$$

Take the square root of 345609.

$$x=\frac{-3±3\sqrt{38401}}{2\times 240}$$

Multiply 2 times 240.

$$x=\frac{-3±3\sqrt{38401}}{480}$$

Now solve the equation x=\frac{-3±3\sqrt{38401}}{480} when ± is plus. Add -3 to 3\sqrt{38401}.

$$x=\frac{3\sqrt{38401}-3}{480}$$

Divide -3+3\sqrt{38401} by 480.

$$x=\frac{\sqrt{38401}-1}{160}$$

Now solve the equation x=\frac{-3±3\sqrt{38401}}{480} when ± is minus. Subtract 3\sqrt{38401} from -3.

$$x=\frac{-3\sqrt{38401}-3}{480}$$

Divide -3-3\sqrt{38401} by 480.

$$x=\frac{-\sqrt{38401}-1}{160}$$

The equation is now solved.

$$x=\frac{\sqrt{38401}-1}{160} x=\frac{-\sqrt{38401}-1}{160}$$

Steps for Completing the Square

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

$$240xx=360+x\left(-3\right)$$

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

$$240x^{2}=360+x\left(-3\right)$$

Subtract x\left(-3\right) from both sides.

$$240x^{2}-x\left(-3\right)=360$$

Multiply -1 and -3 to get 3.

$$240x^{2}+3x=360$$

Divide both sides by 240.

$$\frac{240x^{2}+3x}{240}=\frac{360}{240}$$

Dividing by 240 undoes the multiplication by 240.

$$x^{2}+\frac{3}{240}x=\frac{360}{240}$$

Reduce the fraction \frac{3}{240} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{1}{80}x=\frac{360}{240}$$

Reduce the fraction \frac{360}{240} to lowest terms by extracting and canceling out 120.

$$x^{2}+\frac{1}{80}x=\frac{3}{2}$$

Divide \frac{1}{80}, the coefficient of the x term, by 2 to get \frac{1}{160}. Then add the square of \frac{1}{160} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{80}x+\left(\frac{1}{160}\right)^{2}=\frac{3}{2}+\left(\frac{1}{160}\right)^{2}$$

Square \frac{1}{160} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{80}x+\frac{1}{25600}=\frac{3}{2}+\frac{1}{25600}$$

Add \frac{3}{2} to \frac{1}{25600} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{80}x+\frac{1}{25600}=\frac{38401}{25600}$$

Factor x^{2}+\frac{1}{80}x+\frac{1}{25600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{160}\right)^{2}=\frac{38401}{25600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{160}\right)^{2}}=\sqrt{\frac{38401}{25600}}$$

Simplify.

$$x+\frac{1}{160}=\frac{\sqrt{38401}}{160} x+\frac{1}{160}=-\frac{\sqrt{38401}}{160}$$

Subtract \frac{1}{160} from both sides of the equation.

$$x=\frac{\sqrt{38401}-1}{160} x=\frac{-\sqrt{38401}-1}{160}$$