Solve 1/2-x-1=1/x-2-5-x/3x^2-12

$$\frac{ 1 }{ 2-x } -1 = \frac{ 1 }{ x-2 } - \frac{ 5-x }{ 3 { x }^{ 2 } -12 }$$

Solve for x

$$x=\frac{\sqrt{109}-7}{6}\approx 0.573384418$$

$$x=\frac{-\sqrt{109}-7}{6}\approx -2.906717751$$

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.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(5-x\right)$$

Multiply 3 and -1 to get -3.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

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

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

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

$$-6-3x-3x^{2}+12=3x+6-\left(5-x\right)$$

Add -6 and 12 to get 6.

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

To find the opposite of 5-x, find the opposite of each term.

$$6-3x-3x^{2}=3x+6-5+x$$

Subtract 5 from 6 to get 1.

$$6-3x-3x^{2}=3x+1+x$$

Combine 3x and x to get 4x.

$$6-3x-3x^{2}=4x+1$$

Subtract 4x from both sides.

$$6-3x-3x^{2}-4x=1$$

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

$$6-7x-3x^{2}=1$$

Subtract 1 from both sides.

$$6-7x-3x^{2}-1=0$$

Subtract 1 from 6 to get 5.

$$5-7x-3x^{2}=0$$

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.

$$-3x^{2}-7x+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -7 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 5}}{2\left(-3\right)}$$

Multiply 12 times 5.

$$x=\frac{-\left(-7\right)±\sqrt{49+60}}{2\left(-3\right)}$$

Add 49 to 60.

$$x=\frac{-\left(-7\right)±\sqrt{109}}{2\left(-3\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{109}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{7±\sqrt{109}}{-6}$$

Now solve the equation x=\frac{7±\sqrt{109}}{-6} when ± is plus. Add 7 to \sqrt{109}.

$$x=\frac{\sqrt{109}+7}{-6}$$

Divide 7+\sqrt{109} by -6.

$$x=\frac{-\sqrt{109}-7}{6}$$

Now solve the equation x=\frac{7±\sqrt{109}}{-6} when ± is minus. Subtract \sqrt{109} from 7.

$$x=\frac{7-\sqrt{109}}{-6}$$

Divide 7-\sqrt{109} by -6.

$$x=\frac{\sqrt{109}-7}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{109}-7}{6} x=\frac{\sqrt{109}-7}{6}$$

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.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(5-x\right)$$

Multiply 3 and -1 to get -3.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

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

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(5-x\right)$$

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

$$-6-3x-3x^{2}+12=3x+6-\left(5-x\right)$$

Add -6 and 12 to get 6.

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

To find the opposite of 5-x, find the opposite of each term.

$$6-3x-3x^{2}=3x+6-5+x$$

Subtract 5 from 6 to get 1.

$$6-3x-3x^{2}=3x+1+x$$

Combine 3x and x to get 4x.

$$6-3x-3x^{2}=4x+1$$

Subtract 4x from both sides.

$$6-3x-3x^{2}-4x=1$$

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

$$6-7x-3x^{2}=1$$

Subtract 6 from both sides.

$$-7x-3x^{2}=1-6$$

Subtract 6 from 1 to get -5.

$$-7x-3x^{2}=-5$$

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.

$$-3x^{2}-7x=-5$$

Divide both sides by -3.

$$\frac{-3x^{2}-7x}{-3}=-\frac{5}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{5}{-3}$$

Divide -7 by -3.

$$x^{2}+\frac{7}{3}x=-\frac{5}{-3}$$

Divide -5 by -3.

$$x^{2}+\frac{7}{3}x=\frac{5}{3}$$

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.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=\frac{5}{3}+\left(\frac{7}{6}\right)^{2}$$

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

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{5}{3}+\frac{49}{36}$$

Add \frac{5}{3} to \frac{49}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{109}{36}$$

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}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{109}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{109}{36}}$$

Simplify.

$$x+\frac{7}{6}=\frac{\sqrt{109}}{6} x+\frac{7}{6}=-\frac{\sqrt{109}}{6}$$

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

$$x=\frac{\sqrt{109}-7}{6} x=\frac{-\sqrt{109}-7}{6}$$