Solve x-2/x^2-x-6=3/2x+4-x/4-x^2

$$\frac { x - 2 } { x ^ { 2 } - x - 6 } = \frac { 3 } { 2 x + 4 } - \frac { x } { 4 - x ^ { 2 } }$$

Solve for x

$$x=1$$

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3.333333333$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-x-6,2x+4,4-x^{2}.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply 2x-4 by x-2 and combine like terms.

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

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

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply x^{2}-5x+6 by 3.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

To find the opposite of 6x-2x^{2}, find the opposite of each term.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

Combine -15x and -6x to get -21x.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Subtract 5x^{2} from both sides.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-8x+8=-21x+18$$

Add 21x to both sides.

$$-3x^{2}-8x+8+21x=18$$

Combine -8x and 21x to get 13x.

$$-3x^{2}+13x+8=18$$

Subtract 18 from both sides.

$$-3x^{2}+13x+8-18=0$$

Subtract 18 from 8 to get -10.

$$-3x^{2}+13x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=13 ab=-3\left(-10\right)=30$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=10 b=3$$

Rewrite -3x^{2}+13x-10 as \left(-3x^{2}+10x\right)+\left(3x-10\right).

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

Factor out -x in -3x^{2}+10x.

$$-x\left(3x-10\right)+3x-10$$

Factor out common term 3x-10 by using distributive property.

$$\left(3x-10\right)\left(-x+1\right)$$

To find equation solutions, solve 3x-10=0 and -x+1=0.

$$x=\frac{10}{3} x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-x-6,2x+4,4-x^{2}.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply 2x-4 by x-2 and combine like terms.

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

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

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply x^{2}-5x+6 by 3.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

To find the opposite of 6x-2x^{2}, find the opposite of each term.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

Combine -15x and -6x to get -21x.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Subtract 5x^{2} from both sides.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-8x+8=-21x+18$$

Add 21x to both sides.

$$-3x^{2}-8x+8+21x=18$$

Combine -8x and 21x to get 13x.

$$-3x^{2}+13x+8=18$$

Subtract 18 from both sides.

$$-3x^{2}+13x+8-18=0$$

Subtract 18 from 8 to get -10.

$$-3x^{2}+13x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 13 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-3\right)\left(-10\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-13±\sqrt{169+12\left(-10\right)}}{2\left(-3\right)}$$

Multiply 12 times -10.

$$x=\frac{-13±\sqrt{169-120}}{2\left(-3\right)}$$

Add 169 to -120.

$$x=\frac{-13±\sqrt{49}}{2\left(-3\right)}$$

Take the square root of 49.

$$x=\frac{-13±7}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-13±7}{-6}$$

Now solve the equation x=\frac{-13±7}{-6} when ± is plus. Add -13 to 7.

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

Divide -6 by -6.

$$x=1$$

Now solve the equation x=\frac{-13±7}{-6} when ± is minus. Subtract 7 from -13.

$$x=-\frac{20}{-6}$$

Reduce the fraction \frac{-20}{-6} to lowest terms by extracting and canceling out 2.

$$x=\frac{10}{3}$$

The equation is now solved.

$$x=1 x=\frac{10}{3}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-x-6,2x+4,4-x^{2}.

$$\left(2x-4\right)\left(x-2\right)=\left(x-3\right)\left(x-2\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply 2x-4 by x-2 and combine like terms.

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

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

$$2x^{2}-8x+8=\left(x^{2}-5x+6\right)\times 3-\left(6-2x\right)x$$

Use the distributive property to multiply x^{2}-5x+6 by 3.

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6-2x\right)x$$

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

$$2x^{2}-8x+8=3x^{2}-15x+18-\left(6x-2x^{2}\right)$$

To find the opposite of 6x-2x^{2}, find the opposite of each term.

$$2x^{2}-8x+8=3x^{2}-15x+18-6x+2x^{2}$$

Combine -15x and -6x to get -21x.

$$2x^{2}-8x+8=3x^{2}-21x+18+2x^{2}$$

Combine 3x^{2} and 2x^{2} to get 5x^{2}.

$$2x^{2}-8x+8=5x^{2}-21x+18$$

Subtract 5x^{2} from both sides.

$$2x^{2}-8x+8-5x^{2}=-21x+18$$

Combine 2x^{2} and -5x^{2} to get -3x^{2}.

$$-3x^{2}-8x+8=-21x+18$$

Add 21x to both sides.

$$-3x^{2}-8x+8+21x=18$$

Combine -8x and 21x to get 13x.

$$-3x^{2}+13x+8=18$$

Subtract 8 from both sides.

$$-3x^{2}+13x=18-8$$

Subtract 8 from 18 to get 10.

$$-3x^{2}+13x=10$$

Divide both sides by -3.

$$\frac{-3x^{2}+13x}{-3}=\frac{10}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{13}{-3}x=\frac{10}{-3}$$

Divide 13 by -3.

$$x^{2}-\frac{13}{3}x=\frac{10}{-3}$$

Divide 10 by -3.

$$x^{2}-\frac{13}{3}x=-\frac{10}{3}$$

Divide -\frac{13}{3}, the coefficient of the x term, by 2 to get -\frac{13}{6}. Then add the square of -\frac{13}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{3}x+\left(-\frac{13}{6}\right)^{2}=-\frac{10}{3}+\left(-\frac{13}{6}\right)^{2}$$

Square -\frac{13}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=-\frac{10}{3}+\frac{169}{36}$$

Add -\frac{10}{3} to \frac{169}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{3}x+\frac{169}{36}=\frac{49}{36}$$

Factor x^{2}-\frac{13}{3}x+\frac{169}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x-\frac{13}{6}=\frac{7}{6} x-\frac{13}{6}=-\frac{7}{6}$$

Add \frac{13}{6} to both sides of the equation.

$$x=\frac{10}{3} x=1$$