Solve 5/x+4+4/x-4=2

$$\frac{ 5 }{ x+4 } + \frac{ 4 }{ x-4 } =2$$

Solve for x

$$x = \frac{\sqrt{305} + 9}{4} \approx 6.616062299$$

$$x=\frac{9-\sqrt{305}}{4}\approx -2.116062299$$

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.

$$\left(x-4\right)\times 5+\left(x+4\right)\times 4=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 5.

$$5x-20+\left(x+4\right)\times 4=2\left(x-4\right)\left(x+4\right)$$

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

$$5x-20+4x+16=2\left(x-4\right)\left(x+4\right)$$

Combine 5x and 4x to get 9x.

$$9x-20+16=2\left(x-4\right)\left(x+4\right)$$

Add -20 and 16 to get -4.

$$9x-4=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 2 by x-4.

$$9x-4=\left(2x-8\right)\left(x+4\right)$$

Use the distributive property to multiply 2x-8 by x+4 and combine like terms.

$$9x-4=2x^{2}-32$$

Subtract 2x^{2} from both sides.

$$9x-4-2x^{2}=-32$$

Add 32 to both sides.

$$9x-4-2x^{2}+32=0$$

Add -4 and 32 to get 28.

$$9x+28-2x^{2}=0$$

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.

$$-2x^{2}+9x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 9 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-9±\sqrt{81+8\times 28}}{2\left(-2\right)}$$

Multiply 8 times 28.

$$x=\frac{-9±\sqrt{81+224}}{2\left(-2\right)}$$

Add 81 to 224.

$$x=\frac{-9±\sqrt{305}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-9±\sqrt{305}}{-4}$$

Now solve the equation x=\frac{-9±\sqrt{305}}{-4} when ± is plus. Add -9 to \sqrt{305}.

$$x=\frac{\sqrt{305}-9}{-4}$$

Divide -9+\sqrt{305} by -4.

$$x=\frac{9-\sqrt{305}}{4}$$

Now solve the equation x=\frac{-9±\sqrt{305}}{-4} when ± is minus. Subtract \sqrt{305} from -9.

$$x=\frac{-\sqrt{305}-9}{-4}$$

Divide -9-\sqrt{305} by -4.

$$x=\frac{\sqrt{305}+9}{4}$$

The equation is now solved.

$$x=\frac{9-\sqrt{305}}{4} x=\frac{\sqrt{305}+9}{4}$$

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.

$$\left(x-4\right)\times 5+\left(x+4\right)\times 4=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply x-4 by 5.

$$5x-20+\left(x+4\right)\times 4=2\left(x-4\right)\left(x+4\right)$$

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

$$5x-20+4x+16=2\left(x-4\right)\left(x+4\right)$$

Combine 5x and 4x to get 9x.

$$9x-20+16=2\left(x-4\right)\left(x+4\right)$$

Add -20 and 16 to get -4.

$$9x-4=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 2 by x-4.

$$9x-4=\left(2x-8\right)\left(x+4\right)$$

Use the distributive property to multiply 2x-8 by x+4 and combine like terms.

$$9x-4=2x^{2}-32$$

Subtract 2x^{2} from both sides.

$$9x-4-2x^{2}=-32$$

Add 4 to both sides.

$$9x-2x^{2}=-32+4$$

Add -32 and 4 to get -28.

$$9x-2x^{2}=-28$$

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.

$$-2x^{2}+9x=-28$$

Divide both sides by -2.

$$\frac{-2x^{2}+9x}{-2}=-\frac{28}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{9}{-2}x=-\frac{28}{-2}$$

Divide 9 by -2.

$$x^{2}-\frac{9}{2}x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x^{2}-\frac{9}{2}x=14$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=14+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=14+\frac{81}{16}$$

Add 14 to \frac{81}{16}.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{305}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{305}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{305}{16}}$$

Simplify.

$$x-\frac{9}{4}=\frac{\sqrt{305}}{4} x-\frac{9}{4}=-\frac{\sqrt{305}}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$x=\frac{\sqrt{305}+9}{4} x=\frac{9-\sqrt{305}}{4}$$