Solve 165/x+4+165/x-4=13

$$\frac { 165 } { x + 4 } + \frac { 165 } { x - 4 } = 13$$

Solve for x

$$x=26$$

$$x=-\frac{8}{13}\approx -0.615384615$$

Steps Using the Quadratic Formula

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

$$\left(x-4\right)\times 165+\left(x+4\right)\times 165=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 165.

$$165x-660+\left(x+4\right)\times 165=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 165.

$$165x-660+165x+660=13\left(x-4\right)\left(x+4\right)$$

Combine 165x and 165x to get 330x.

$$330x-660+660=13\left(x-4\right)\left(x+4\right)$$

Add -660 and 660 to get 0.

$$330x=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 13 by x-4.

$$330x=\left(13x-52\right)\left(x+4\right)$$

Use the distributive property to multiply 13x-52 by x+4 and combine like terms.

$$330x=13x^{2}-208$$

Subtract 13x^{2} from both sides.

$$330x-13x^{2}=-208$$

Add 208 to both sides.

$$330x-13x^{2}+208=0$$

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.

$$-13x^{2}+330x+208=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, 330 for b, and 208 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-330±\sqrt{330^{2}-4\left(-13\right)\times 208}}{2\left(-13\right)}$$

Square 330.

$$x=\frac{-330±\sqrt{108900-4\left(-13\right)\times 208}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$x=\frac{-330±\sqrt{108900+52\times 208}}{2\left(-13\right)}$$

Multiply 52 times 208.

$$x=\frac{-330±\sqrt{108900+10816}}{2\left(-13\right)}$$

Add 108900 to 10816.

$$x=\frac{-330±\sqrt{119716}}{2\left(-13\right)}$$

Take the square root of 119716.

$$x=\frac{-330±346}{2\left(-13\right)}$$

Multiply 2 times -13.

$$x=\frac{-330±346}{-26}$$

Now solve the equation x=\frac{-330±346}{-26} when ± is plus. Add -330 to 346.

$$x=\frac{16}{-26}$$

Reduce the fraction \frac{16}{-26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{8}{13}$$

Now solve the equation x=\frac{-330±346}{-26} when ± is minus. Subtract 346 from -330.

$$x=-\frac{676}{-26}$$

Divide -676 by -26.

$$x=26$$

The equation is now solved.

$$x=-\frac{8}{13} x=26$$

Steps for Completing the Square

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

$$\left(x-4\right)\times 165+\left(x+4\right)\times 165=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 165.

$$165x-660+\left(x+4\right)\times 165=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x+4 by 165.

$$165x-660+165x+660=13\left(x-4\right)\left(x+4\right)$$

Combine 165x and 165x to get 330x.

$$330x-660+660=13\left(x-4\right)\left(x+4\right)$$

Add -660 and 660 to get 0.

$$330x=13\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 13 by x-4.

$$330x=\left(13x-52\right)\left(x+4\right)$$

Use the distributive property to multiply 13x-52 by x+4 and combine like terms.

$$330x=13x^{2}-208$$

Subtract 13x^{2} from both sides.

$$330x-13x^{2}=-208$$

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.

$$-13x^{2}+330x=-208$$

Divide both sides by -13.

$$\frac{-13x^{2}+330x}{-13}=-\frac{208}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$x^{2}+\frac{330}{-13}x=-\frac{208}{-13}$$

Divide 330 by -13.

$$x^{2}-\frac{330}{13}x=-\frac{208}{-13}$$

Divide -208 by -13.

$$x^{2}-\frac{330}{13}x=16$$

Divide -\frac{330}{13}, the coefficient of the x term, by 2 to get -\frac{165}{13}. Then add the square of -\frac{165}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{330}{13}x+\left(-\frac{165}{13}\right)^{2}=16+\left(-\frac{165}{13}\right)^{2}$$

Square -\frac{165}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{330}{13}x+\frac{27225}{169}=16+\frac{27225}{169}$$

Add 16 to \frac{27225}{169}.

$$x^{2}-\frac{330}{13}x+\frac{27225}{169}=\frac{29929}{169}$$

Factor x^{2}-\frac{330}{13}x+\frac{27225}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{165}{13}\right)^{2}=\frac{29929}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{165}{13}\right)^{2}}=\sqrt{\frac{29929}{169}}$$

Simplify.

$$x-\frac{165}{13}=\frac{173}{13} x-\frac{165}{13}=-\frac{173}{13}$$

Add \frac{165}{13} to both sides of the equation.

$$x=26 x=-\frac{8}{13}$$