Solve 16/x^2-9+4/x^2-5x+6-5left(x+2right)/6-x-x^2=0

$$\frac{ 16 }{ { x }^{ 2 } -9 } + \frac{ 4 }{ { x }^{ 2 } -5x+6 } - \frac{ 5 \left( x+2 \right) }{ 6-x- { x }^{ 2 } } = 0$$

Solve for x

$$x=-5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply x-2 by 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Combine 16x and 4x to get 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Add -32 and 12 to get -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply 3-x by 5.

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Use the distributive property to multiply 15-5x by x+2 and combine like terms.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

To find the opposite of 5x+30-5x^{2}, find the opposite of each term.

$$20x-20-5x-30+5x^{2}=0$$

Combine 20x and -5x to get 15x.

$$15x-20-30+5x^{2}=0$$

Subtract 30 from -20 to get -50.

$$15x-50+5x^{2}=0$$

Divide both sides by 5.

$$3x-10+x^{2}=0$$

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

$$x^{2}+3x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=3 ab=1\left(-10\right)=-10$$

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 -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=-2 b=5$$

Rewrite x^{2}+3x-10 as \left(x^{2}-2x\right)+\left(5x-10\right).

$$\left(x^{2}-2x\right)+\left(5x-10\right)$$

Factor out x in the first and 5 in the second group.

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

Factor out common term x-2 by using distributive property.

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

To find equation solutions, solve x-2=0 and x+5=0.

$$x=2 x=-5$$

Variable x cannot be equal to 2.

$$x=-5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply x-2 by 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Combine 16x and 4x to get 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Add -32 and 12 to get -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply 3-x by 5.

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Use the distributive property to multiply 15-5x by x+2 and combine like terms.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

To find the opposite of 5x+30-5x^{2}, find the opposite of each term.

$$20x-20-5x-30+5x^{2}=0$$

Combine 20x and -5x to get 15x.

$$15x-20-30+5x^{2}=0$$

Subtract 30 from -20 to get -50.

$$15x-50+5x^{2}=0$$

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.

$$5x^{2}+15x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 15 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5\left(-50\right)}}{2\times 5}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 5\left(-50\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-15±\sqrt{225-20\left(-50\right)}}{2\times 5}$$

Multiply -20 times -50.

$$x=\frac{-15±\sqrt{225+1000}}{2\times 5}$$

Add 225 to 1000.

$$x=\frac{-15±\sqrt{1225}}{2\times 5}$$

Take the square root of 1225.

$$x=\frac{-15±35}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-15±35}{10}$$

Now solve the equation x=\frac{-15±35}{10} when ± is plus. Add -15 to 35.

$$x=\frac{20}{10}$$

Divide 20 by 10.

$$x=2$$

Now solve the equation x=\frac{-15±35}{10} when ± is minus. Subtract 35 from -15.

$$x=-\frac{50}{10}$$

Divide -50 by 10.

$$x=-5$$

The equation is now solved.

$$x=2 x=-5$$

Variable x cannot be equal to 2.

$$x=-5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+3\right), the least common multiple of x^{2}-9,x^{2}-5x+6,6-x-x^{2}.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply x-2 by 16.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Combine 16x and 4x to get 20x.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

Add -32 and 12 to get -20.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

Use the distributive property to multiply 3-x by 5.

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

Use the distributive property to multiply 15-5x by x+2 and combine like terms.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

To find the opposite of 5x+30-5x^{2}, find the opposite of each term.

$$20x-20-5x-30+5x^{2}=0$$

Combine 20x and -5x to get 15x.

$$15x-20-30+5x^{2}=0$$

Subtract 30 from -20 to get -50.

$$15x-50+5x^{2}=0$$

Add 50 to both sides. Anything plus zero gives itself.

$$15x+5x^{2}=50$$

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.

$$5x^{2}+15x=50$$

Divide both sides by 5.

$$\frac{5x^{2}+15x}{5}=\frac{50}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{15}{5}x=\frac{50}{5}$$

Divide 15 by 5.

$$x^{2}+3x=\frac{50}{5}$$

Divide 50 by 5.

$$x^{2}+3x=10$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

Add 10 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

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

Subtract \frac{3}{2} from both sides of the equation.

$$x=2 x=-5$$

Variable x cannot be equal to 2.

$$x=-5$$