Solve 120/x+120/x+3=2

$$\frac{ 120 }{ x } + \frac{ 120 }{ x+3 } =2$$

Solve for x

$$x = \frac{3 \sqrt{1601} + 117}{2} \approx 118.518747071$$

$$x=\frac{117-3\sqrt{1601}}{2}\approx -1.518747071$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x,x+3.

$$\left(x+3\right)\times 120+x\times 120=2x\left(x+3\right)$$

Use the distributive property to multiply x+3 by 120.

$$120x+360+x\times 120=2x\left(x+3\right)$$

Combine 120x and x\times 120 to get 240x.

$$240x+360=2x\left(x+3\right)$$

Use the distributive property to multiply 2x by x+3.

$$240x+360=2x^{2}+6x$$

Subtract 2x^{2} from both sides.

$$240x+360-2x^{2}=6x$$

Subtract 6x from both sides.

$$240x+360-2x^{2}-6x=0$$

Combine 240x and -6x to get 234x.

$$234x+360-2x^{2}=0$$

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.

$$-2x^{2}+234x+360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 234 for b, and 360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-234±\sqrt{234^{2}-4\left(-2\right)\times 360}}{2\left(-2\right)}$$

Square 234.

$$x=\frac{-234±\sqrt{54756-4\left(-2\right)\times 360}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-234±\sqrt{54756+8\times 360}}{2\left(-2\right)}$$

Multiply 8 times 360.

$$x=\frac{-234±\sqrt{54756+2880}}{2\left(-2\right)}$$

Add 54756 to 2880.

$$x=\frac{-234±\sqrt{57636}}{2\left(-2\right)}$$

Take the square root of 57636.

$$x=\frac{-234±6\sqrt{1601}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-234±6\sqrt{1601}}{-4}$$

Now solve the equation x=\frac{-234±6\sqrt{1601}}{-4} when ± is plus. Add -234 to 6\sqrt{1601}.

$$x=\frac{6\sqrt{1601}-234}{-4}$$

Divide -234+6\sqrt{1601} by -4.

$$x=\frac{117-3\sqrt{1601}}{2}$$

Now solve the equation x=\frac{-234±6\sqrt{1601}}{-4} when ± is minus. Subtract 6\sqrt{1601} from -234.

$$x=\frac{-6\sqrt{1601}-234}{-4}$$

Divide -234-6\sqrt{1601} by -4.

$$x=\frac{3\sqrt{1601}+117}{2}$$

The equation is now solved.

$$x=\frac{117-3\sqrt{1601}}{2} x=\frac{3\sqrt{1601}+117}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+3\right), the least common multiple of x,x+3.

$$\left(x+3\right)\times 120+x\times 120=2x\left(x+3\right)$$

Use the distributive property to multiply x+3 by 120.

$$120x+360+x\times 120=2x\left(x+3\right)$$

Combine 120x and x\times 120 to get 240x.

$$240x+360=2x\left(x+3\right)$$

Use the distributive property to multiply 2x by x+3.

$$240x+360=2x^{2}+6x$$

Subtract 2x^{2} from both sides.

$$240x+360-2x^{2}=6x$$

Subtract 6x from both sides.

$$240x+360-2x^{2}-6x=0$$

Combine 240x and -6x to get 234x.

$$234x+360-2x^{2}=0$$

Subtract 360 from both sides. Anything subtracted from zero gives its negation.

$$234x-2x^{2}=-360$$

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.

$$-2x^{2}+234x=-360$$

Divide both sides by -2.

$$\frac{-2x^{2}+234x}{-2}=-\frac{360}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{234}{-2}x=-\frac{360}{-2}$$

Divide 234 by -2.

$$x^{2}-117x=-\frac{360}{-2}$$

Divide -360 by -2.

$$x^{2}-117x=180$$

Divide -117, the coefficient of the x term, by 2 to get -\frac{117}{2}. Then add the square of -\frac{117}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-117x+\left(-\frac{117}{2}\right)^{2}=180+\left(-\frac{117}{2}\right)^{2}$$

Square -\frac{117}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-117x+\frac{13689}{4}=180+\frac{13689}{4}$$

Add 180 to \frac{13689}{4}.

$$x^{2}-117x+\frac{13689}{4}=\frac{14409}{4}$$

Factor x^{2}-117x+\frac{13689}{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}.

$$\left(x-\frac{117}{2}\right)^{2}=\frac{14409}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{117}{2}\right)^{2}}=\sqrt{\frac{14409}{4}}$$

Simplify.

$$x-\frac{117}{2}=\frac{3\sqrt{1601}}{2} x-\frac{117}{2}=-\frac{3\sqrt{1601}}{2}$$

Add \frac{117}{2} to both sides of the equation.

$$x=\frac{3\sqrt{1601}+117}{2} x=\frac{117-3\sqrt{1601}}{2}$$