Solve 15/x-9+15/x+9=375/100

$$\frac { 15 } { x - 9 } + \frac { 15 } { x + 9 } = \frac { 375 } { 100 }$$

Solve for x

$$x=\sqrt{97}+4\approx 13.848857802$$

$$x=4-\sqrt{97}\approx -5.848857802$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by 100\left(x-9\right)\left(x+9\right), the least common multiple of x-9,x+9,100.

$$\left(100x+900\right)\times 15+\left(100x-900\right)\times 15=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply 100x+900 by 15.

$$1500x+13500+\left(100x-900\right)\times 15=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply 100x-900 by 15.

$$1500x+13500+1500x-13500=\left(x^{2}-81\right)\times 375$$

Combine 1500x and 1500x to get 3000x.

$$3000x+13500-13500=\left(x^{2}-81\right)\times 375$$

Subtract 13500 from 13500 to get 0.

$$3000x=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply x^{2}-81 by 375.

$$3000x=375x^{2}-30375$$

Subtract 375x^{2} from both sides.

$$3000x-375x^{2}=-30375$$

Add 30375 to both sides.

$$3000x-375x^{2}+30375=0$$

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.

$$-375x^{2}+3000x+30375=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -375 for a, 3000 for b, and 30375 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3000±\sqrt{3000^{2}-4\left(-375\right)\times 30375}}{2\left(-375\right)}$$

Square 3000.

$$x=\frac{-3000±\sqrt{9000000-4\left(-375\right)\times 30375}}{2\left(-375\right)}$$

Multiply -4 times -375.

$$x=\frac{-3000±\sqrt{9000000+1500\times 30375}}{2\left(-375\right)}$$

Multiply 1500 times 30375.

$$x=\frac{-3000±\sqrt{9000000+45562500}}{2\left(-375\right)}$$

Add 9000000 to 45562500.

$$x=\frac{-3000±\sqrt{54562500}}{2\left(-375\right)}$$

Take the square root of 54562500.

$$x=\frac{-3000±750\sqrt{97}}{2\left(-375\right)}$$

Multiply 2 times -375.

$$x=\frac{-3000±750\sqrt{97}}{-750}$$

Now solve the equation x=\frac{-3000±750\sqrt{97}}{-750} when ± is plus. Add -3000 to 750\sqrt{97}.

$$x=\frac{750\sqrt{97}-3000}{-750}$$

Divide -3000+750\sqrt{97} by -750.

$$x=4-\sqrt{97}$$

Now solve the equation x=\frac{-3000±750\sqrt{97}}{-750} when ± is minus. Subtract 750\sqrt{97} from -3000.

$$x=\frac{-750\sqrt{97}-3000}{-750}$$

Divide -3000-750\sqrt{97} by -750.

$$x=\sqrt{97}+4$$

The equation is now solved.

$$x=4-\sqrt{97} x=\sqrt{97}+4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by 100\left(x-9\right)\left(x+9\right), the least common multiple of x-9,x+9,100.

$$\left(100x+900\right)\times 15+\left(100x-900\right)\times 15=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply 100x+900 by 15.

$$1500x+13500+\left(100x-900\right)\times 15=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply 100x-900 by 15.

$$1500x+13500+1500x-13500=\left(x^{2}-81\right)\times 375$$

Combine 1500x and 1500x to get 3000x.

$$3000x+13500-13500=\left(x^{2}-81\right)\times 375$$

Subtract 13500 from 13500 to get 0.

$$3000x=\left(x^{2}-81\right)\times 375$$

Use the distributive property to multiply x^{2}-81 by 375.

$$3000x=375x^{2}-30375$$

Subtract 375x^{2} from both sides.

$$3000x-375x^{2}=-30375$$

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.

$$-375x^{2}+3000x=-30375$$

Divide both sides by -375.

$$\frac{-375x^{2}+3000x}{-375}=-\frac{30375}{-375}$$

Dividing by -375 undoes the multiplication by -375.

$$x^{2}+\frac{3000}{-375}x=-\frac{30375}{-375}$$

Divide 3000 by -375.

$$x^{2}-8x=-\frac{30375}{-375}$$

Divide -30375 by -375.

$$x^{2}-8x=81$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=81+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=81+16$$

Add 81 to 16.

$$x^{2}-8x+16=97$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=97$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{97}$$

Simplify.

$$x-4=\sqrt{97} x-4=-\sqrt{97}$$

Add 4 to both sides of the equation.

$$x=\sqrt{97}+4 x=4-\sqrt{97}$$