Solve 60/x+10+60/x-10=8

$$\frac { 60 } { x + 10 } + \frac { 60 } { x - 10 } = 8$$

Solve for x

$$x=-5$$

$$x=20$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,10 since division by zero is not defined. Multiply both sides of the equation by \left(x-10\right)\left(x+10\right), the least common multiple of x+10,x-10.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply x-10 by 60.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply x+10 by 60.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

Combine 60x and 60x to get 120x.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

Add -600 and 600 to get 0.

$$120x=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply 8 by x-10.

$$120x=\left(8x-80\right)\left(x+10\right)$$

Use the distributive property to multiply 8x-80 by x+10 and combine like terms.

$$120x=8x^{2}-800$$

Subtract 8x^{2} from both sides.

$$120x-8x^{2}=-800$$

Add 800 to both sides.

$$120x-8x^{2}+800=0$$

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.

$$-8x^{2}+120x+800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 120 for b, and 800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\times 800}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-120±\sqrt{14400+32\times 800}}{2\left(-8\right)}$$

Multiply 32 times 800.

$$x=\frac{-120±\sqrt{14400+25600}}{2\left(-8\right)}$$

Add 14400 to 25600.

$$x=\frac{-120±\sqrt{40000}}{2\left(-8\right)}$$

Take the square root of 40000.

$$x=\frac{-120±200}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-120±200}{-16}$$

Now solve the equation x=\frac{-120±200}{-16} when ± is plus. Add -120 to 200.

$$x=\frac{80}{-16}$$

Divide 80 by -16.

$$x=-5$$

Now solve the equation x=\frac{-120±200}{-16} when ± is minus. Subtract 200 from -120.

$$x=-\frac{320}{-16}$$

Divide -320 by -16.

$$x=20$$

The equation is now solved.

$$x=-5 x=20$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,10 since division by zero is not defined. Multiply both sides of the equation by \left(x-10\right)\left(x+10\right), the least common multiple of x+10,x-10.

$$\left(x-10\right)\times 60+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply x-10 by 60.

$$60x-600+\left(x+10\right)\times 60=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply x+10 by 60.

$$60x-600+60x+600=8\left(x-10\right)\left(x+10\right)$$

Combine 60x and 60x to get 120x.

$$120x-600+600=8\left(x-10\right)\left(x+10\right)$$

Add -600 and 600 to get 0.

$$120x=8\left(x-10\right)\left(x+10\right)$$

Use the distributive property to multiply 8 by x-10.

$$120x=\left(8x-80\right)\left(x+10\right)$$

Use the distributive property to multiply 8x-80 by x+10 and combine like terms.

$$120x=8x^{2}-800$$

Subtract 8x^{2} from both sides.

$$120x-8x^{2}=-800$$

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.

$$-8x^{2}+120x=-800$$

Divide both sides by -8.

$$\frac{-8x^{2}+120x}{-8}=-\frac{800}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{120}{-8}x=-\frac{800}{-8}$$

Divide 120 by -8.

$$x^{2}-15x=-\frac{800}{-8}$$

Divide -800 by -8.

$$x^{2}-15x=100$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=100+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=100+\frac{225}{4}$$

Add 100 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{625}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{625}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{625}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{25}{2} x-\frac{15}{2}=-\frac{25}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=20 x=-5$$