Solve x+3/x+9+7/x-9=7/x-9

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

Solve for x

$$x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x+9\right), the least common multiple of x+9,x-9.

$$\left(x-9\right)\left(x+3\right)+\left(x+9\right)\times 7=\left(x+9\right)\times 7$$

Use the distributive property to multiply x-9 by x+3 and combine like terms.

$$x^{2}-6x-27+\left(x+9\right)\times 7=\left(x+9\right)\times 7$$

Use the distributive property to multiply x+9 by 7.

$$x^{2}-6x-27+7x+63=\left(x+9\right)\times 7$$

Combine -6x and 7x to get x.

$$x^{2}+x-27+63=\left(x+9\right)\times 7$$

Add -27 and 63 to get 36.

$$x^{2}+x+36=\left(x+9\right)\times 7$$

Use the distributive property to multiply x+9 by 7.

$$x^{2}+x+36=7x+63$$

Subtract 7x from both sides.

$$x^{2}+x+36-7x=63$$

Combine x and -7x to get -6x.

$$x^{2}-6x+36=63$$

Subtract 63 from both sides.

$$x^{2}-6x+36-63=0$$

Subtract 63 from 36 to get -27.

$$x^{2}-6x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-27\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-27\right)}}{2}$$

Multiply -4 times -27.

$$x=\frac{-\left(-6\right)±\sqrt{36+108}}{2}$$

Add 36 to 108.

$$x=\frac{-\left(-6\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-\left(-6\right)±12}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±12}{2}$$

Now solve the equation x=\frac{6±12}{2} when ± is plus. Add 6 to 12.

$$x=\frac{18}{2}$$

Divide 18 by 2.

$$x=9$$

Now solve the equation x=\frac{6±12}{2} when ± is minus. Subtract 12 from 6.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=9 x=-3$$

Variable x cannot be equal to 9.

$$x=-3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -9,9 since division by zero is not defined. Multiply both sides of the equation by \left(x-9\right)\left(x+9\right), the least common multiple of x+9,x-9.

$$\left(x-9\right)\left(x+3\right)+\left(x+9\right)\times 7=\left(x+9\right)\times 7$$

Use the distributive property to multiply x-9 by x+3 and combine like terms.

$$x^{2}-6x-27+\left(x+9\right)\times 7=\left(x+9\right)\times 7$$

Use the distributive property to multiply x+9 by 7.

$$x^{2}-6x-27+7x+63=\left(x+9\right)\times 7$$

Combine -6x and 7x to get x.

$$x^{2}+x-27+63=\left(x+9\right)\times 7$$

Add -27 and 63 to get 36.

$$x^{2}+x+36=\left(x+9\right)\times 7$$

Use the distributive property to multiply x+9 by 7.

$$x^{2}+x+36=7x+63$$

Subtract 7x from both sides.

$$x^{2}+x+36-7x=63$$

Combine x and -7x to get -6x.

$$x^{2}-6x+36=63$$

Subtract 36 from both sides.

$$x^{2}-6x=63-36$$

Subtract 36 from 63 to get 27.

$$x^{2}-6x=27$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=27+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=27+9$$

Add 27 to 9.

$$x^{2}-6x+9=36$$

Factor x^{2}-6x+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}.

$$\left(x-3\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-3=6 x-3=-6$$

Add 3 to both sides of the equation.

$$x=9 x=-3$$

Variable x cannot be equal to 9.

$$x=-3$$