Solve 2-x/x+2=x+3/3-x+6

$$\frac{ 2-x }{ x+2 } = \frac{ x+3 }{ 3-x } +6$$

Solve for x

$$x = \frac{\sqrt{70} + 4}{3} \approx 4.122200088$$

$$x=\frac{4-\sqrt{70}}{3}\approx -1.455533422$$

Steps Using the Quadratic Formula

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

$$\left(x-3\right)\left(2-x\right)=-\left(2+x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply x-3 by 2-x and combine like terms.

$$5x-x^{2}-6=-\left(2+x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply -1 by 2+x.

$$5x-x^{2}-6=\left(-2-x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply -2-x by x+3 and combine like terms.

$$5x-x^{2}-6=-5x-6-x^{2}+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply x-3 by x+2 and combine like terms.

$$5x-x^{2}-6=-5x-6-x^{2}+\left(x^{2}-x-6\right)\times 6$$

Use the distributive property to multiply x^{2}-x-6 by 6.

$$5x-x^{2}-6=-5x-6-x^{2}+6x^{2}-6x-36$$

Combine -x^{2} and 6x^{2} to get 5x^{2}.

$$5x-x^{2}-6=-5x-6+5x^{2}-6x-36$$

Combine -5x and -6x to get -11x.

$$5x-x^{2}-6=-11x-6+5x^{2}-36$$

Subtract 36 from -6 to get -42.

$$5x-x^{2}-6=-11x-42+5x^{2}$$

Add 11x to both sides.

$$5x-x^{2}-6+11x=-42+5x^{2}$$

Combine 5x and 11x to get 16x.

$$16x-x^{2}-6=-42+5x^{2}$$

Subtract -42 from both sides.

$$16x-x^{2}-6-\left(-42\right)=5x^{2}$$

The opposite of -42 is 42.

$$16x-x^{2}-6+42=5x^{2}$$

Subtract 5x^{2} from both sides.

$$16x-x^{2}-6+42-5x^{2}=0$$

Add -6 and 42 to get 36.

$$16x-x^{2}+36-5x^{2}=0$$

Combine -x^{2} and -5x^{2} to get -6x^{2}.

$$16x-6x^{2}+36=0$$

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.

$$-6x^{2}+16x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 16 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-6\right)\times 36}}{2\left(-6\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-6\right)\times 36}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-16±\sqrt{256+24\times 36}}{2\left(-6\right)}$$

Multiply 24 times 36.

$$x=\frac{-16±\sqrt{256+864}}{2\left(-6\right)}$$

Add 256 to 864.

$$x=\frac{-16±\sqrt{1120}}{2\left(-6\right)}$$

Take the square root of 1120.

$$x=\frac{-16±4\sqrt{70}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-16±4\sqrt{70}}{-12}$$

Now solve the equation x=\frac{-16±4\sqrt{70}}{-12} when ± is plus. Add -16 to 4\sqrt{70}.

$$x=\frac{4\sqrt{70}-16}{-12}$$

Divide -16+4\sqrt{70} by -12.

$$x=\frac{4-\sqrt{70}}{3}$$

Now solve the equation x=\frac{-16±4\sqrt{70}}{-12} when ± is minus. Subtract 4\sqrt{70} from -16.

$$x=\frac{-4\sqrt{70}-16}{-12}$$

Divide -16-4\sqrt{70} by -12.

$$x=\frac{\sqrt{70}+4}{3}$$

The equation is now solved.

$$x=\frac{4-\sqrt{70}}{3} x=\frac{\sqrt{70}+4}{3}$$

Steps for Completing the Square

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

$$\left(x-3\right)\left(2-x\right)=-\left(2+x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply x-3 by 2-x and combine like terms.

$$5x-x^{2}-6=-\left(2+x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply -1 by 2+x.

$$5x-x^{2}-6=\left(-2-x\right)\left(x+3\right)+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply -2-x by x+3 and combine like terms.

$$5x-x^{2}-6=-5x-6-x^{2}+\left(x-3\right)\left(x+2\right)\times 6$$

Use the distributive property to multiply x-3 by x+2 and combine like terms.

$$5x-x^{2}-6=-5x-6-x^{2}+\left(x^{2}-x-6\right)\times 6$$

Use the distributive property to multiply x^{2}-x-6 by 6.

$$5x-x^{2}-6=-5x-6-x^{2}+6x^{2}-6x-36$$

Combine -x^{2} and 6x^{2} to get 5x^{2}.

$$5x-x^{2}-6=-5x-6+5x^{2}-6x-36$$

Combine -5x and -6x to get -11x.

$$5x-x^{2}-6=-11x-6+5x^{2}-36$$

Subtract 36 from -6 to get -42.

$$5x-x^{2}-6=-11x-42+5x^{2}$$

Add 11x to both sides.

$$5x-x^{2}-6+11x=-42+5x^{2}$$

Combine 5x and 11x to get 16x.

$$16x-x^{2}-6=-42+5x^{2}$$

Subtract 5x^{2} from both sides.

$$16x-x^{2}-6-5x^{2}=-42$$

Combine -x^{2} and -5x^{2} to get -6x^{2}.

$$16x-6x^{2}-6=-42$$

Add 6 to both sides.

$$16x-6x^{2}=-42+6$$

Add -42 and 6 to get -36.

$$16x-6x^{2}=-36$$

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.

$$-6x^{2}+16x=-36$$

Divide both sides by -6.

$$\frac{-6x^{2}+16x}{-6}=-\frac{36}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{16}{-6}x=-\frac{36}{-6}$$

Reduce the fraction \frac{16}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{8}{3}x=-\frac{36}{-6}$$

Divide -36 by -6.

$$x^{2}-\frac{8}{3}x=6$$

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.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=6+\left(-\frac{4}{3}\right)^{2}$$

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

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=6+\frac{16}{9}$$

Add 6 to \frac{16}{9}.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{70}{9}$$

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}.

$$\left(x-\frac{4}{3}\right)^{2}=\frac{70}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{\frac{70}{9}}$$

Simplify.

$$x-\frac{4}{3}=\frac{\sqrt{70}}{3} x-\frac{4}{3}=-\frac{\sqrt{70}}{3}$$

Add \frac{4}{3} to both sides of the equation.

$$x=\frac{\sqrt{70}+4}{3} x=\frac{4-\sqrt{70}}{3}$$