Solve 720/x+30+720/x-30=10

$$\frac { 720 } { x + 30 } + \frac { 720 } { x - 30 } = 10$$

Solve for x

$$x=-6$$

$$x=150$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -30,30 since division by zero is not defined. Multiply both sides of the equation by \left(x-30\right)\left(x+30\right), the least common multiple of x+30,x-30.

$$\left(x-30\right)\times 720+\left(x+30\right)\times 720=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply x-30 by 720.

$$720x-21600+\left(x+30\right)\times 720=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply x+30 by 720.

$$720x-21600+720x+21600=10\left(x-30\right)\left(x+30\right)$$

Combine 720x and 720x to get 1440x.

$$1440x-21600+21600=10\left(x-30\right)\left(x+30\right)$$

Add -21600 and 21600 to get 0.

$$1440x=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply 10 by x-30.

$$1440x=\left(10x-300\right)\left(x+30\right)$$

Use the distributive property to multiply 10x-300 by x+30 and combine like terms.

$$1440x=10x^{2}-9000$$

Subtract 10x^{2} from both sides.

$$1440x-10x^{2}=-9000$$

Add 9000 to both sides.

$$1440x-10x^{2}+9000=0$$

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.

$$-10x^{2}+1440x+9000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1440 for b, and 9000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1440±\sqrt{1440^{2}-4\left(-10\right)\times 9000}}{2\left(-10\right)}$$

Square 1440.

$$x=\frac{-1440±\sqrt{2073600-4\left(-10\right)\times 9000}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-1440±\sqrt{2073600+40\times 9000}}{2\left(-10\right)}$$

Multiply 40 times 9000.

$$x=\frac{-1440±\sqrt{2073600+360000}}{2\left(-10\right)}$$

Add 2073600 to 360000.

$$x=\frac{-1440±\sqrt{2433600}}{2\left(-10\right)}$$

Take the square root of 2433600.

$$x=\frac{-1440±1560}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-1440±1560}{-20}$$

Now solve the equation x=\frac{-1440±1560}{-20} when ± is plus. Add -1440 to 1560.

$$x=\frac{120}{-20}$$

Divide 120 by -20.

$$x=-6$$

Now solve the equation x=\frac{-1440±1560}{-20} when ± is minus. Subtract 1560 from -1440.

$$x=-\frac{3000}{-20}$$

Divide -3000 by -20.

$$x=150$$

The equation is now solved.

$$x=-6 x=150$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -30,30 since division by zero is not defined. Multiply both sides of the equation by \left(x-30\right)\left(x+30\right), the least common multiple of x+30,x-30.

$$\left(x-30\right)\times 720+\left(x+30\right)\times 720=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply x-30 by 720.

$$720x-21600+\left(x+30\right)\times 720=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply x+30 by 720.

$$720x-21600+720x+21600=10\left(x-30\right)\left(x+30\right)$$

Combine 720x and 720x to get 1440x.

$$1440x-21600+21600=10\left(x-30\right)\left(x+30\right)$$

Add -21600 and 21600 to get 0.

$$1440x=10\left(x-30\right)\left(x+30\right)$$

Use the distributive property to multiply 10 by x-30.

$$1440x=\left(10x-300\right)\left(x+30\right)$$

Use the distributive property to multiply 10x-300 by x+30 and combine like terms.

$$1440x=10x^{2}-9000$$

Subtract 10x^{2} from both sides.

$$1440x-10x^{2}=-9000$$

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.

$$-10x^{2}+1440x=-9000$$

Divide both sides by -10.

$$\frac{-10x^{2}+1440x}{-10}=-\frac{9000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{1440}{-10}x=-\frac{9000}{-10}$$

Divide 1440 by -10.

$$x^{2}-144x=-\frac{9000}{-10}$$

Divide -9000 by -10.

$$x^{2}-144x=900$$

Divide -144, the coefficient of the x term, by 2 to get -72. Then add the square of -72 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-144x+\left(-72\right)^{2}=900+\left(-72\right)^{2}$$

Square -72.

$$x^{2}-144x+5184=900+5184$$

Add 900 to 5184.

$$x^{2}-144x+5184=6084$$

Factor x^{2}-144x+5184. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-72\right)^{2}=6084$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-72\right)^{2}}=\sqrt{6084}$$

Simplify.

$$x-72=78 x-72=-78$$

Add 72 to both sides of the equation.

$$x=150 x=-6$$