Solve 30/x^2+5x+6-x/x+2=2x+1/x+3

$$\frac { 30 } { x ^ { 2 } + 5 x + 6 } - \frac { x } { x + 2 } = \frac { 2 x + 1 } { x + 3 }$$

Solve for x

$$x = -\frac{14}{3} = -4\frac{2}{3} \approx -4.666666667$$

$$x=2$$

Steps Using Factoring By Grouping

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

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

To find the opposite of x^{2}+3x, find the opposite of each term.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

Use the distributive property to multiply x+2 by 2x+1 and combine like terms.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Subtract 2x^{2} from both sides.

$$30-x^{2}-3x-2x^{2}=5x+2$$

Combine -x^{2} and -2x^{2} to get -3x^{2}.

$$30-3x^{2}-3x=5x+2$$

Subtract 5x from both sides.

$$30-3x^{2}-3x-5x=2$$

Combine -3x and -5x to get -8x.

$$30-3x^{2}-8x=2$$

Subtract 2 from both sides.

$$30-3x^{2}-8x-2=0$$

Subtract 2 from 30 to get 28.

$$28-3x^{2}-8x=0$$

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

$$-3x^{2}-8x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-3\times 28=-84$$

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 -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -8.

$$a=6 b=-14$$

Rewrite -3x^{2}-8x+28 as \left(-3x^{2}+6x\right)+\left(-14x+28\right).

$$\left(-3x^{2}+6x\right)+\left(-14x+28\right)$$

Factor out 3x in the first and 14 in the second group.

$$3x\left(-x+2\right)+14\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(3x+14\right)$$

To find equation solutions, solve -x+2=0 and 3x+14=0.

$$x=2 x=-\frac{14}{3}$$

Steps Using the Quadratic Formula

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

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

To find the opposite of x^{2}+3x, find the opposite of each term.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

Use the distributive property to multiply x+2 by 2x+1 and combine like terms.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Subtract 2x^{2} from both sides.

$$30-x^{2}-3x-2x^{2}=5x+2$$

Combine -x^{2} and -2x^{2} to get -3x^{2}.

$$30-3x^{2}-3x=5x+2$$

Subtract 5x from both sides.

$$30-3x^{2}-3x-5x=2$$

Combine -3x and -5x to get -8x.

$$30-3x^{2}-8x=2$$

Subtract 2 from both sides.

$$30-3x^{2}-8x-2=0$$

Subtract 2 from 30 to get 28.

$$28-3x^{2}-8x=0$$

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.

$$-3x^{2}-8x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -8 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-8\right)±\sqrt{64+12\times 28}}{2\left(-3\right)}$$

Multiply 12 times 28.

$$x=\frac{-\left(-8\right)±\sqrt{64+336}}{2\left(-3\right)}$$

Add 64 to 336.

$$x=\frac{-\left(-8\right)±\sqrt{400}}{2\left(-3\right)}$$

Take the square root of 400.

$$x=\frac{-\left(-8\right)±20}{2\left(-3\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±20}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{8±20}{-6}$$

Now solve the equation x=\frac{8±20}{-6} when ± is plus. Add 8 to 20.

$$x=\frac{28}{-6}$$

Reduce the fraction \frac{28}{-6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{14}{3}$$

Now solve the equation x=\frac{8±20}{-6} when ± is minus. Subtract 20 from 8.

$$x=-\frac{12}{-6}$$

Divide -12 by -6.

$$x=2$$

The equation is now solved.

$$x=-\frac{14}{3} x=2$$

Steps for Completing the Square

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

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

To find the opposite of x^{2}+3x, find the opposite of each term.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

Use the distributive property to multiply x+2 by 2x+1 and combine like terms.

$$30-x^{2}-3x=2x^{2}+5x+2$$

Subtract 2x^{2} from both sides.

$$30-x^{2}-3x-2x^{2}=5x+2$$

Combine -x^{2} and -2x^{2} to get -3x^{2}.

$$30-3x^{2}-3x=5x+2$$

Subtract 5x from both sides.

$$30-3x^{2}-3x-5x=2$$

Combine -3x and -5x to get -8x.

$$30-3x^{2}-8x=2$$

Subtract 30 from both sides.

$$-3x^{2}-8x=2-30$$

Subtract 30 from 2 to get -28.

$$-3x^{2}-8x=-28$$

Divide both sides by -3.

$$\frac{-3x^{2}-8x}{-3}=-\frac{28}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{8}{-3}\right)x=-\frac{28}{-3}$$

Divide -8 by -3.

$$x^{2}+\frac{8}{3}x=-\frac{28}{-3}$$

Divide -28 by -3.

$$x^{2}+\frac{8}{3}x=\frac{28}{3}$$

Divide \frac{8}{3}, the coefficient of the x term, by 2 to get \frac{4}{3}. Then add the square of \frac{4}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{28}{3}+\left(\frac{4}{3}\right)^{2}$$

Square \frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{28}{3}+\frac{16}{9}$$

Add \frac{28}{3} to \frac{16}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{100}{9}$$

Factor x^{2}+\frac{8}{3}x+\frac{16}{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}.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{100}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Simplify.

$$x+\frac{4}{3}=\frac{10}{3} x+\frac{4}{3}=-\frac{10}{3}$$

Subtract \frac{4}{3} from both sides of the equation.

$$x=2 x=-\frac{14}{3}$$