Solve x-3/2x-12=1-2/6-x-1/x^2-12x+36

$$\frac { x - 3 } { 2 x - 12 } = 1 - \frac { 2 } { 6 - x } - \frac { 1 } { x ^ { 2 } - 12 x + 36 }$$

Solve for x

$$x=4$$

$$x=7$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-6\right)^{2}, the least common multiple of 2x-12,6-x,x^{2}-12x+36.

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

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

$$x^{2}-9x+18=2\left(x-6\right)^{2}-\left(12-2x\right)\times 2-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-9x+18=2\left(x^{2}-12x+36\right)-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 2 by x^{2}-12x+36.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 12-2x by 2.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(24-4x\right)-2$$

To find the opposite of 24-4x, find the opposite of each term.

$$x^{2}-9x+18=2x^{2}-24x+72-24+4x-2$$

Subtract 24 from 72 to get 48.

$$x^{2}-9x+18=2x^{2}-24x+48+4x-2$$

Combine -24x and 4x to get -20x.

$$x^{2}-9x+18=2x^{2}-20x+48-2$$

Subtract 2 from 48 to get 46.

$$x^{2}-9x+18=2x^{2}-20x+46$$

Subtract 2x^{2} from both sides.

$$x^{2}-9x+18-2x^{2}=-20x+46$$

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

$$-x^{2}-9x+18=-20x+46$$

Add 20x to both sides.

$$-x^{2}-9x+18+20x=46$$

Combine -9x and 20x to get 11x.

$$-x^{2}+11x+18=46$$

Subtract 46 from both sides.

$$-x^{2}+11x+18-46=0$$

Subtract 46 from 18 to get -28.

$$-x^{2}+11x-28=0$$

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

$$a+b=11 ab=-\left(-28\right)=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 11.

$$a=7 b=4$$

Rewrite -x^{2}+11x-28 as \left(-x^{2}+7x\right)+\left(4x-28\right).

$$\left(-x^{2}+7x\right)+\left(4x-28\right)$$

Factor out -x in the first and 4 in the second group.

$$-x\left(x-7\right)+4\left(x-7\right)$$

Factor out common term x-7 by using distributive property.

$$\left(x-7\right)\left(-x+4\right)$$

To find equation solutions, solve x-7=0 and -x+4=0.

$$x=7 x=4$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-6\right)^{2}, the least common multiple of 2x-12,6-x,x^{2}-12x+36.

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

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

$$x^{2}-9x+18=2\left(x-6\right)^{2}-\left(12-2x\right)\times 2-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-9x+18=2\left(x^{2}-12x+36\right)-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 2 by x^{2}-12x+36.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 12-2x by 2.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(24-4x\right)-2$$

To find the opposite of 24-4x, find the opposite of each term.

$$x^{2}-9x+18=2x^{2}-24x+72-24+4x-2$$

Subtract 24 from 72 to get 48.

$$x^{2}-9x+18=2x^{2}-24x+48+4x-2$$

Combine -24x and 4x to get -20x.

$$x^{2}-9x+18=2x^{2}-20x+48-2$$

Subtract 2 from 48 to get 46.

$$x^{2}-9x+18=2x^{2}-20x+46$$

Subtract 2x^{2} from both sides.

$$x^{2}-9x+18-2x^{2}=-20x+46$$

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

$$-x^{2}-9x+18=-20x+46$$

Add 20x to both sides.

$$-x^{2}-9x+18+20x=46$$

Combine -9x and 20x to get 11x.

$$-x^{2}+11x+18=46$$

Subtract 46 from both sides.

$$-x^{2}+11x+18-46=0$$

Subtract 46 from 18 to get -28.

$$-x^{2}+11x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$x=\frac{-11±\sqrt{121-112}}{2\left(-1\right)}$$

Add 121 to -112.

$$x=\frac{-11±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$x=\frac{-11±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±3}{-2}$$

Now solve the equation x=\frac{-11±3}{-2} when ± is plus. Add -11 to 3.

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

Now solve the equation x=\frac{-11±3}{-2} when ± is minus. Subtract 3 from -11.

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

Divide -14 by -2.

$$x=7$$

The equation is now solved.

$$x=4 x=7$$

Steps for Completing the Square

Variable x cannot be equal to 6 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-6\right)^{2}, the least common multiple of 2x-12,6-x,x^{2}-12x+36.

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

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

$$x^{2}-9x+18=2\left(x-6\right)^{2}-\left(12-2x\right)\times 2-2$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-9x+18=2\left(x^{2}-12x+36\right)-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 2 by x^{2}-12x+36.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(12-2x\right)\times 2-2$$

Use the distributive property to multiply 12-2x by 2.

$$x^{2}-9x+18=2x^{2}-24x+72-\left(24-4x\right)-2$$

To find the opposite of 24-4x, find the opposite of each term.

$$x^{2}-9x+18=2x^{2}-24x+72-24+4x-2$$

Subtract 24 from 72 to get 48.

$$x^{2}-9x+18=2x^{2}-24x+48+4x-2$$

Combine -24x and 4x to get -20x.

$$x^{2}-9x+18=2x^{2}-20x+48-2$$

Subtract 2 from 48 to get 46.

$$x^{2}-9x+18=2x^{2}-20x+46$$

Subtract 2x^{2} from both sides.

$$x^{2}-9x+18-2x^{2}=-20x+46$$

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

$$-x^{2}-9x+18=-20x+46$$

Add 20x to both sides.

$$-x^{2}-9x+18+20x=46$$

Combine -9x and 20x to get 11x.

$$-x^{2}+11x+18=46$$

Subtract 18 from both sides.

$$-x^{2}+11x=46-18$$

Subtract 18 from 46 to get 28.

$$-x^{2}+11x=28$$

Divide both sides by -1.

$$\frac{-x^{2}+11x}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{11}{-1}x=\frac{28}{-1}$$

Divide 11 by -1.

$$x^{2}-11x=\frac{28}{-1}$$

Divide 28 by -1.

$$x^{2}-11x=-28$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-28+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=-28+\frac{121}{4}$$

Add -28 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{9}{4}$$

Factor x^{2}-11x+\frac{121}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{3}{2} x-\frac{11}{2}=-\frac{3}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=7 x=4$$