Solve 140/x+140/x-4=4

$$\frac { 140 } { x } + \frac { 140 } { x - 4 } = 4$$

Solve for x

$$x=\sqrt{1229}+37\approx 72.057096286$$

$$x=37-\sqrt{1229}\approx 1.942903714$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x,x-4.

$$\left(x-4\right)\times 140+x\times 140=4x\left(x-4\right)$$

Use the distributive property to multiply x-4 by 140.

$$140x-560+x\times 140=4x\left(x-4\right)$$

Combine 140x and x\times 140 to get 280x.

$$280x-560=4x\left(x-4\right)$$

Use the distributive property to multiply 4x by x-4.

$$280x-560=4x^{2}-16x$$

Subtract 4x^{2} from both sides.

$$280x-560-4x^{2}=-16x$$

Add 16x to both sides.

$$280x-560-4x^{2}+16x=0$$

Combine 280x and 16x to get 296x.

$$296x-560-4x^{2}=0$$

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.

$$-4x^{2}+296x-560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 296 for b, and -560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-296±\sqrt{296^{2}-4\left(-4\right)\left(-560\right)}}{2\left(-4\right)}$$

Square 296.

$$x=\frac{-296±\sqrt{87616-4\left(-4\right)\left(-560\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-296±\sqrt{87616+16\left(-560\right)}}{2\left(-4\right)}$$

Multiply 16 times -560.

$$x=\frac{-296±\sqrt{87616-8960}}{2\left(-4\right)}$$

Add 87616 to -8960.

$$x=\frac{-296±\sqrt{78656}}{2\left(-4\right)}$$

Take the square root of 78656.

$$x=\frac{-296±8\sqrt{1229}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-296±8\sqrt{1229}}{-8}$$

Now solve the equation x=\frac{-296±8\sqrt{1229}}{-8} when ± is plus. Add -296 to 8\sqrt{1229}.

$$x=\frac{8\sqrt{1229}-296}{-8}$$

Divide -296+8\sqrt{1229} by -8.

$$x=37-\sqrt{1229}$$

Now solve the equation x=\frac{-296±8\sqrt{1229}}{-8} when ± is minus. Subtract 8\sqrt{1229} from -296.

$$x=\frac{-8\sqrt{1229}-296}{-8}$$

Divide -296-8\sqrt{1229} by -8.

$$x=\sqrt{1229}+37$$

The equation is now solved.

$$x=37-\sqrt{1229} x=\sqrt{1229}+37$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,4 since division by zero is not defined. Multiply both sides of the equation by x\left(x-4\right), the least common multiple of x,x-4.

$$\left(x-4\right)\times 140+x\times 140=4x\left(x-4\right)$$

Use the distributive property to multiply x-4 by 140.

$$140x-560+x\times 140=4x\left(x-4\right)$$

Combine 140x and x\times 140 to get 280x.

$$280x-560=4x\left(x-4\right)$$

Use the distributive property to multiply 4x by x-4.

$$280x-560=4x^{2}-16x$$

Subtract 4x^{2} from both sides.

$$280x-560-4x^{2}=-16x$$

Add 16x to both sides.

$$280x-560-4x^{2}+16x=0$$

Combine 280x and 16x to get 296x.

$$296x-560-4x^{2}=0$$

Add 560 to both sides. Anything plus zero gives itself.

$$296x-4x^{2}=560$$

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.

$$-4x^{2}+296x=560$$

Divide both sides by -4.

$$\frac{-4x^{2}+296x}{-4}=\frac{560}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{296}{-4}x=\frac{560}{-4}$$

Divide 296 by -4.

$$x^{2}-74x=\frac{560}{-4}$$

Divide 560 by -4.

$$x^{2}-74x=-140$$

Divide -74, the coefficient of the x term, by 2 to get -37. Then add the square of -37 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-74x+\left(-37\right)^{2}=-140+\left(-37\right)^{2}$$

Square -37.

$$x^{2}-74x+1369=-140+1369$$

Add -140 to 1369.

$$x^{2}-74x+1369=1229$$

Factor x^{2}-74x+1369. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-37\right)^{2}=1229$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-37\right)^{2}}=\sqrt{1229}$$

Simplify.

$$x-37=\sqrt{1229} x-37=-\sqrt{1229}$$

Add 37 to both sides of the equation.

$$x=\sqrt{1229}+37 x=37-\sqrt{1229}$$