Solve 8/x+5-32/x^2-25=2-2/x-5

$$\frac { 8 } { x + 5 } - \frac { 32 } { x ^ { 2 } - 25 } = 2 - \frac { 2 } { x - 5 }$$

Solve for x

$$x=2$$

$$x=3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x+5,x^{2}-25,x-5.

$$\left(x-5\right)\times 8-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by 8.

$$8x-40-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Subtract 32 from -40 to get -72.

$$8x-72=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by x+5 and combine like terms.

$$8x-72=\left(x^{2}-25\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x^{2}-25 by 2.

$$8x-72=2x^{2}-50-\left(x+5\right)\times 2$$

Use the distributive property to multiply x+5 by 2.

$$8x-72=2x^{2}-50-\left(2x+10\right)$$

To find the opposite of 2x+10, find the opposite of each term.

$$8x-72=2x^{2}-50-2x-10$$

Subtract 10 from -50 to get -60.

$$8x-72=2x^{2}-60-2x$$

Subtract 2x^{2} from both sides.

$$8x-72-2x^{2}=-60-2x$$

Subtract -60 from both sides.

$$8x-72-2x^{2}-\left(-60\right)=-2x$$

The opposite of -60 is 60.

$$8x-72-2x^{2}+60=-2x$$

Add 2x to both sides.

$$8x-72-2x^{2}+60+2x=0$$

Add -72 and 60 to get -12.

$$8x-12-2x^{2}+2x=0$$

Combine 8x and 2x to get 10x.

$$10x-12-2x^{2}=0$$

Divide both sides by 2.

$$5x-6-x^{2}=0$$

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

$$-x^{2}+5x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=5 ab=-\left(-6\right)=6$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=3 b=2$$

Rewrite -x^{2}+5x-6 as \left(-x^{2}+3x\right)+\left(2x-6\right).

$$\left(-x^{2}+3x\right)+\left(2x-6\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-3\right)+2\left(x-3\right)$$

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

$$\left(x-3\right)\left(-x+2\right)$$

To find equation solutions, solve x-3=0 and -x+2=0.

$$x=3 x=2$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x+5,x^{2}-25,x-5.

$$\left(x-5\right)\times 8-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by 8.

$$8x-40-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Subtract 32 from -40 to get -72.

$$8x-72=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by x+5 and combine like terms.

$$8x-72=\left(x^{2}-25\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x^{2}-25 by 2.

$$8x-72=2x^{2}-50-\left(x+5\right)\times 2$$

Use the distributive property to multiply x+5 by 2.

$$8x-72=2x^{2}-50-\left(2x+10\right)$$

To find the opposite of 2x+10, find the opposite of each term.

$$8x-72=2x^{2}-50-2x-10$$

Subtract 10 from -50 to get -60.

$$8x-72=2x^{2}-60-2x$$

Subtract 2x^{2} from both sides.

$$8x-72-2x^{2}=-60-2x$$

Subtract -60 from both sides.

$$8x-72-2x^{2}-\left(-60\right)=-2x$$

The opposite of -60 is 60.

$$8x-72-2x^{2}+60=-2x$$

Add 2x to both sides.

$$8x-72-2x^{2}+60+2x=0$$

Add -72 and 60 to get -12.

$$8x-12-2x^{2}+2x=0$$

Combine 8x and 2x to get 10x.

$$10x-12-2x^{2}=0$$

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.

$$-2x^{2}+10x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-10±\sqrt{100+8\left(-12\right)}}{2\left(-2\right)}$$

Multiply 8 times -12.

$$x=\frac{-10±\sqrt{100-96}}{2\left(-2\right)}$$

Add 100 to -96.

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

Take the square root of 4.

$$x=\frac{-10±2}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-10±2}{-4}$$

Now solve the equation x=\frac{-10±2}{-4} when ± is plus. Add -10 to 2.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

Now solve the equation x=\frac{-10±2}{-4} when ± is minus. Subtract 2 from -10.

$$x=-\frac{12}{-4}$$

Divide -12 by -4.

$$x=3$$

The equation is now solved.

$$x=2 x=3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)\left(x+5\right), the least common multiple of x+5,x^{2}-25,x-5.

$$\left(x-5\right)\times 8-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by 8.

$$8x-40-32=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Subtract 32 from -40 to get -72.

$$8x-72=\left(x-5\right)\left(x+5\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x-5 by x+5 and combine like terms.

$$8x-72=\left(x^{2}-25\right)\times 2-\left(x+5\right)\times 2$$

Use the distributive property to multiply x^{2}-25 by 2.

$$8x-72=2x^{2}-50-\left(x+5\right)\times 2$$

Use the distributive property to multiply x+5 by 2.

$$8x-72=2x^{2}-50-\left(2x+10\right)$$

To find the opposite of 2x+10, find the opposite of each term.

$$8x-72=2x^{2}-50-2x-10$$

Subtract 10 from -50 to get -60.

$$8x-72=2x^{2}-60-2x$$

Subtract 2x^{2} from both sides.

$$8x-72-2x^{2}=-60-2x$$

Add 2x to both sides.

$$8x-72-2x^{2}+2x=-60$$

Combine 8x and 2x to get 10x.

$$10x-72-2x^{2}=-60$$

Add 72 to both sides.

$$10x-2x^{2}=-60+72$$

Add -60 and 72 to get 12.

$$10x-2x^{2}=12$$

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.

$$-2x^{2}+10x=12$$

Divide both sides by -2.

$$\frac{-2x^{2}+10x}{-2}=\frac{12}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{10}{-2}x=\frac{12}{-2}$$

Divide 10 by -2.

$$x^{2}-5x=\frac{12}{-2}$$

Divide 12 by -2.

$$x^{2}-5x=-6$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-6+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-6+\frac{25}{4}$$

Add -6 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{1}{2} x-\frac{5}{2}=-\frac{1}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=3 x=2$$