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

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

Solve for x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x-2.

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

Use the distributive property to multiply x+2 by x.

$$5+x^{2}+2x=4\left(x-2\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$5+x^{2}+2x-4x^{2}=-16$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$5-3x^{2}+2x=-16$$

Add 16 to both sides.

$$5-3x^{2}+2x+16=0$$

Add 5 and 16 to get 21.

$$21-3x^{2}+2x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-3x^{2}+2x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=2 ab=-3\times 21=-63$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -63.

$$-1,63 -3,21 -7,9$$

Calculate the sum for each pair.

$$-1+63=62 -3+21=18 -7+9=2$$

The solution is the pair that gives sum 2.

$$a=9 b=-7$$

Rewrite -3x^{2}+2x+21 as \left(-3x^{2}+9x\right)+\left(-7x+21\right).

$$\left(-3x^{2}+9x\right)+\left(-7x+21\right)$$

Factor out 3x in the first and 7 in the second group.

$$3x\left(-x+3\right)+7\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(3x+7\right)$$

To find equation solutions, solve -x+3=0 and 3x+7=0.

$$x=3 x=-\frac{7}{3}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x-2.

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

Use the distributive property to multiply x+2 by x.

$$5+x^{2}+2x=4\left(x-2\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$5+x^{2}+2x-4x^{2}=-16$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$5-3x^{2}+2x=-16$$

Add 16 to both sides.

$$5-3x^{2}+2x+16=0$$

Add 5 and 16 to get 21.

$$21-3x^{2}+2x=0$$

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.

$$-3x^{2}+2x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 2 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\times 21}}{2\left(-3\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)\times 21}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12\times 21}}{2\left(-3\right)}$$

Multiply 12 times 21.

$$x=\frac{-2±\sqrt{4+252}}{2\left(-3\right)}$$

Add 4 to 252.

$$x=\frac{-2±\sqrt{256}}{2\left(-3\right)}$$

Take the square root of 256.

$$x=\frac{-2±16}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±16}{-6}$$

Now solve the equation x=\frac{-2±16}{-6} when ± is plus. Add -2 to 16.

$$x=\frac{14}{-6}$$

Reduce the fraction \frac{14}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{3}$$

Now solve the equation x=\frac{-2±16}{-6} when ± is minus. Subtract 16 from -2.

$$x=-\frac{18}{-6}$$

Divide -18 by -6.

$$x=3$$

The equation is now solved.

$$x=-\frac{7}{3} x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x-2.

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

Use the distributive property to multiply x+2 by x.

$$5+x^{2}+2x=4\left(x-2\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

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

$$5+x^{2}+2x=4x^{2}-16$$

Subtract 4x^{2} from both sides.

$$5+x^{2}+2x-4x^{2}=-16$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$5-3x^{2}+2x=-16$$

Subtract 5 from both sides.

$$-3x^{2}+2x=-16-5$$

Subtract 5 from -16 to get -21.

$$-3x^{2}+2x=-21$$

Divide both sides by -3.

$$\frac{-3x^{2}+2x}{-3}=-\frac{21}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{2}{-3}x=-\frac{21}{-3}$$

Divide 2 by -3.

$$x^{2}-\frac{2}{3}x=-\frac{21}{-3}$$

Divide -21 by -3.

$$x^{2}-\frac{2}{3}x=7$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=7+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=7+\frac{1}{9}$$

Add 7 to \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{64}{9}$$

Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{64}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Simplify.

$$x-\frac{1}{3}=\frac{8}{3} x-\frac{1}{3}=-\frac{8}{3}$$

Add \frac{1}{3} to both sides of the equation.

$$x=3 x=-\frac{7}{3}$$