Solve x/x-5-9/x+5=50/x^2-25

$$\frac { x } { x - 5 } - \frac { 9 } { x + 5 } = \frac { 50 } { x ^ { 2 } - 25 }$$

Solve for x

$$x=-1$$

Steps Using Factoring

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+5,x^{2}-25.

$$\left(x+5\right)x-\left(x-5\right)\times 9=50$$

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

$$x^{2}+5x-\left(x-5\right)\times 9=50$$

Use the distributive property to multiply x-5 by 9.

$$x^{2}+5x-\left(9x-45\right)=50$$

To find the opposite of 9x-45, find the opposite of each term.

$$x^{2}+5x-9x+45=50$$

Combine 5x and -9x to get -4x.

$$x^{2}-4x+45=50$$

Subtract 50 from both sides.

$$x^{2}-4x+45-50=0$$

Subtract 50 from 45 to get -5.

$$x^{2}-4x-5=0$$

To solve the equation, factor x^{2}-4x-5 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-4 ab=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-5\right)\left(x+1\right)$$

To find equation solutions, solve x-5=0 and x+1=0.

$$x=5 x=-1$$

Variable x cannot be equal to 5.

$$x=-1$$

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+5,x^{2}-25.

$$\left(x+5\right)x-\left(x-5\right)\times 9=50$$

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

$$x^{2}+5x-\left(x-5\right)\times 9=50$$

Use the distributive property to multiply x-5 by 9.

$$x^{2}+5x-\left(9x-45\right)=50$$

To find the opposite of 9x-45, find the opposite of each term.

$$x^{2}+5x-9x+45=50$$

Combine 5x and -9x to get -4x.

$$x^{2}-4x+45=50$$

Subtract 50 from both sides.

$$x^{2}-4x+45-50=0$$

Subtract 50 from 45 to get -5.

$$x^{2}-4x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-4 ab=1\left(-5\right)=-5$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=-5 b=1$$

Rewrite x^{2}-4x-5 as \left(x^{2}-5x\right)+\left(x-5\right).

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

Factor out x in x^{2}-5x.

$$x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(x+1\right)$$

To find equation solutions, solve x-5=0 and x+1=0.

$$x=5 x=-1$$

Variable x cannot be equal to 5.

$$x=-1$$

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+5,x^{2}-25.

$$\left(x+5\right)x-\left(x-5\right)\times 9=50$$

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

$$x^{2}+5x-\left(x-5\right)\times 9=50$$

Use the distributive property to multiply x-5 by 9.

$$x^{2}+5x-\left(9x-45\right)=50$$

To find the opposite of 9x-45, find the opposite of each term.

$$x^{2}+5x-9x+45=50$$

Combine 5x and -9x to get -4x.

$$x^{2}-4x+45=50$$

Subtract 50 from both sides.

$$x^{2}-4x+45-50=0$$

Subtract 50 from 45 to get -5.

$$x^{2}-4x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-5\right)}}{2}$$

Multiply -4 times -5.

$$x=\frac{-\left(-4\right)±\sqrt{16+20}}{2}$$

Add 16 to 20.

$$x=\frac{-\left(-4\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-\left(-4\right)±6}{2}$$

The opposite of -4 is 4.

$$x=\frac{4±6}{2}$$

Now solve the equation x=\frac{4±6}{2} when ± is plus. Add 4 to 6.

$$x=\frac{10}{2}$$

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{4±6}{2} when ± is minus. Subtract 6 from 4.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=5 x=-1$$

Variable x cannot be equal to 5.

$$x=-1$$

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+5,x^{2}-25.

$$\left(x+5\right)x-\left(x-5\right)\times 9=50$$

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

$$x^{2}+5x-\left(x-5\right)\times 9=50$$

Use the distributive property to multiply x-5 by 9.

$$x^{2}+5x-\left(9x-45\right)=50$$

To find the opposite of 9x-45, find the opposite of each term.

$$x^{2}+5x-9x+45=50$$

Combine 5x and -9x to get -4x.

$$x^{2}-4x+45=50$$

Subtract 45 from both sides.

$$x^{2}-4x=50-45$$

Subtract 45 from 50 to get 5.

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

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

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

Square -2.

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

Add 5 to 4.

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

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-2=3 x-2=-3$$

Add 2 to both sides of the equation.

$$x=5 x=-1$$

Variable x cannot be equal to 5.

$$x=-1$$