Solve x-2/x+3-30/x^2-9=4/9

$$\frac{ x-2 }{ x+3 } - \frac{ 30 }{ { x }^{ 2 } -9 } = \frac{ 4 }{ 9 }$$

Solve for x

$$x=12$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 9\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x^{2}-9,9.

$$\left(9x-27\right)\left(x-2\right)-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 9x-27 by x-2 and combine like terms.

$$9x^{2}-45x+54-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Multiply -9 and 30 to get -270.

$$9x^{2}-45x+54-270=4\left(x-3\right)\left(x+3\right)$$

Subtract 270 from 54 to get -216.

$$9x^{2}-45x-216=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 4 by x-3.

$$9x^{2}-45x-216=\left(4x-12\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-12 by x+3 and combine like terms.

$$9x^{2}-45x-216=4x^{2}-36$$

Subtract 4x^{2} from both sides.

$$9x^{2}-45x-216-4x^{2}=-36$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}-45x-216=-36$$

Add 36 to both sides.

$$5x^{2}-45x-216+36=0$$

Add -216 and 36 to get -180.

$$5x^{2}-45x-180=0$$

Divide both sides by 5.

$$x^{2}-9x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=-9 ab=1\left(-36\right)=-36$$

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. List all such integer pairs that give product -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -9.

$$a=-12 b=3$$

Rewrite x^{2}-9x-36 as \left(x^{2}-12x\right)+\left(3x-36\right).

$$\left(x^{2}-12x\right)+\left(3x-36\right)$$

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

$$x\left(x-12\right)+3\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+3\right)$$

To find equation solutions, solve x-12=0 and x+3=0.

$$x=12 x=-3$$

Variable x cannot be equal to -3.

$$x=12$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 9\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x^{2}-9,9.

$$\left(9x-27\right)\left(x-2\right)-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 9x-27 by x-2 and combine like terms.

$$9x^{2}-45x+54-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Multiply -9 and 30 to get -270.

$$9x^{2}-45x+54-270=4\left(x-3\right)\left(x+3\right)$$

Subtract 270 from 54 to get -216.

$$9x^{2}-45x-216=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 4 by x-3.

$$9x^{2}-45x-216=\left(4x-12\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-12 by x+3 and combine like terms.

$$9x^{2}-45x-216=4x^{2}-36$$

Subtract 4x^{2} from both sides.

$$9x^{2}-45x-216-4x^{2}=-36$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}-45x-216=-36$$

Add 36 to both sides.

$$5x^{2}-45x-216+36=0$$

Add -216 and 36 to get -180.

$$5x^{2}-45x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -45 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 5\left(-180\right)}}{2\times 5}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\times 5\left(-180\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-45\right)±\sqrt{2025-20\left(-180\right)}}{2\times 5}$$

Multiply -20 times -180.

$$x=\frac{-\left(-45\right)±\sqrt{2025+3600}}{2\times 5}$$

Add 2025 to 3600.

$$x=\frac{-\left(-45\right)±\sqrt{5625}}{2\times 5}$$

Take the square root of 5625.

$$x=\frac{-\left(-45\right)±75}{2\times 5}$$

The opposite of -45 is 45.

$$x=\frac{45±75}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{45±75}{10}$$

Now solve the equation x=\frac{45±75}{10} when ± is plus. Add 45 to 75.

$$x=\frac{120}{10}$$

Divide 120 by 10.

$$x=12$$

Now solve the equation x=\frac{45±75}{10} when ± is minus. Subtract 75 from 45.

$$x=-\frac{30}{10}$$

Divide -30 by 10.

$$x=-3$$

The equation is now solved.

$$x=12 x=-3$$

Variable x cannot be equal to -3.

$$x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by 9\left(x-3\right)\left(x+3\right), the least common multiple of x+3,x^{2}-9,9.

$$\left(9x-27\right)\left(x-2\right)-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 9x-27 by x-2 and combine like terms.

$$9x^{2}-45x+54-9\times 30=4\left(x-3\right)\left(x+3\right)$$

Multiply -9 and 30 to get -270.

$$9x^{2}-45x+54-270=4\left(x-3\right)\left(x+3\right)$$

Subtract 270 from 54 to get -216.

$$9x^{2}-45x-216=4\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 4 by x-3.

$$9x^{2}-45x-216=\left(4x-12\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-12 by x+3 and combine like terms.

$$9x^{2}-45x-216=4x^{2}-36$$

Subtract 4x^{2} from both sides.

$$9x^{2}-45x-216-4x^{2}=-36$$

Combine 9x^{2} and -4x^{2} to get 5x^{2}.

$$5x^{2}-45x-216=-36$$

Add 216 to both sides.

$$5x^{2}-45x=-36+216$$

Add -36 and 216 to get 180.

$$5x^{2}-45x=180$$

Divide both sides by 5.

$$\frac{5x^{2}-45x}{5}=\frac{180}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{45}{5}\right)x=\frac{180}{5}$$

Divide -45 by 5.

$$x^{2}-9x=\frac{180}{5}$$

Divide 180 by 5.

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

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=36+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=36+\frac{81}{4}$$

Add 36 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{225}{4}$$

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

$$\left(x-\frac{9}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{15}{2} x-\frac{9}{2}=-\frac{15}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=12 x=-3$$

Variable x cannot be equal to -3.

$$x=12$$