Solve 1/2-x-1=1/x-2-6-x/3x^2-12

$$\frac{ 1 }{ 2-x } -1 = \frac{ 1 }{ x-2 } - \frac{ 6-x }{ 3 { x }^{ 2 } -12 }$$

Solve for x

$$x=-3$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-2\right)\left(x+2\right), the least common multiple of 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Multiply 3 and -1 to get -3.

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

Use the distributive property to multiply -3 by x-2.

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

Use the distributive property to multiply -3x+6 by x+2 and combine like terms.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Add -6 and 12 to get 6.

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

To find the opposite of 6-x, find the opposite of each term.

$$6-3x-3x^{2}=3x+6-6+x$$

Subtract 6 from 6 to get 0.

$$6-3x-3x^{2}=3x+x$$

Combine 3x and x to get 4x.

$$6-3x-3x^{2}=4x$$

Subtract 4x from both sides.

$$6-3x-3x^{2}-4x=0$$

Combine -3x and -4x to get -7x.

$$6-7x-3x^{2}=0$$

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

$$-3x^{2}-7x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-3\times 6=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -7.

$$a=2 b=-9$$

Rewrite -3x^{2}-7x+6 as \left(-3x^{2}+2x\right)+\left(-9x+6\right).

$$\left(-3x^{2}+2x\right)+\left(-9x+6\right)$$

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

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

Factor out common term 3x-2 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-2\right)\left(x+2\right), the least common multiple of 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Multiply 3 and -1 to get -3.

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

Use the distributive property to multiply -3 by x-2.

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

Use the distributive property to multiply -3x+6 by x+2 and combine like terms.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Add -6 and 12 to get 6.

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

To find the opposite of 6-x, find the opposite of each term.

$$6-3x-3x^{2}=3x+6-6+x$$

Subtract 6 from 6 to get 0.

$$6-3x-3x^{2}=3x+x$$

Combine 3x and x to get 4x.

$$6-3x-3x^{2}=4x$$

Subtract 4x from both sides.

$$6-3x-3x^{2}-4x=0$$

Combine -3x and -4x to get -7x.

$$6-7x-3x^{2}=0$$

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.

$$-3x^{2}-7x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -7 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 6}}{2\left(-3\right)}$$

Multiply 12 times 6.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\left(-3\right)}$$

Add 49 to 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\left(-3\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-7\right)±11}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±11}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±11}{-6}$$

Now solve the equation x=\frac{7±11}{-6} when ± is plus. Add 7 to 11.

$$x=\frac{18}{-6}$$

Divide 18 by -6.

$$x=-3$$

Now solve the equation x=\frac{7±11}{-6} when ± is minus. Subtract 11 from 7.

$$x=-\frac{4}{-6}$$

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-2\right)\left(x+2\right), the least common multiple of 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Multiply 3 and -1 to get -3.

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

Use the distributive property to multiply -3 by x-2.

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

Use the distributive property to multiply -3x+6 by x+2 and combine like terms.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Add -6 and 12 to get 6.

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

To find the opposite of 6-x, find the opposite of each term.

$$6-3x-3x^{2}=3x+6-6+x$$

Subtract 6 from 6 to get 0.

$$6-3x-3x^{2}=3x+x$$

Combine 3x and x to get 4x.

$$6-3x-3x^{2}=4x$$

Subtract 4x from both sides.

$$6-3x-3x^{2}-4x=0$$

Combine -3x and -4x to get -7x.

$$6-7x-3x^{2}=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-7x-3x^{2}=-6$$

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.

$$-3x^{2}-7x=-6$$

Divide both sides by -3.

$$\frac{-3x^{2}-7x}{-3}=-\frac{6}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{6}{-3}$$

Divide -7 by -3.

$$x^{2}+\frac{7}{3}x=-\frac{6}{-3}$$

Divide -6 by -3.

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

Divide \frac{7}{3}, the coefficient of the x term, by 2 to get \frac{7}{6}. Then add the square of \frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}$$

Square \frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=2+\frac{49}{36}$$

Add 2 to \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{121}{36}$$

Factor x^{2}+\frac{7}{3}x+\frac{49}{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}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{121}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{11}{6} x+\frac{7}{6}=-\frac{11}{6}$$

Subtract \frac{7}{6} from both sides of the equation.

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