Solve 2x/x-4+3/x-3+4=30+5x^2-36x/x^2-7x+12

$$\frac { 2 x } { x - 4 } + \frac { 3 } { x - 3 } + 4 = \frac { 30 + 5 x ^ { 2 } - 36 x } { x ^ { 2 } - 7 x + 12 }$$

Solve for x

$$x=-3$$

$$x=-2$$

Steps Using Factoring

Variable x cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Combine -6x and 3x to get -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x^{2}-7x+12 by 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Combine -3x and -28x to get -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Add -12 and 48 to get 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Subtract 30 from both sides.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Subtract 30 from 36 to get 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Subtract 5x^{2} from both sides.

$$6x^{2}-31x+6-5x^{2}=-36x$$

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

$$x^{2}-31x+6=-36x$$

Add 36x to both sides.

$$x^{2}-31x+6+36x=0$$

Combine -31x and 36x to get 5x.

$$x^{2}+5x+6=0$$

To solve the equation, factor x^{2}+5x+6 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=5 ab=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=2 b=3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

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

$$x=-2 x=-3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Combine -6x and 3x to get -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x^{2}-7x+12 by 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Combine -3x and -28x to get -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Add -12 and 48 to get 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Subtract 30 from both sides.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Subtract 30 from 36 to get 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Subtract 5x^{2} from both sides.

$$6x^{2}-31x+6-5x^{2}=-36x$$

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

$$x^{2}-31x+6=-36x$$

Add 36x to both sides.

$$x^{2}-31x+6+36x=0$$

Combine -31x and 36x to get 5x.

$$x^{2}+5x+6=0$$

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

$$a+b=5 ab=1\times 6=6$$

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 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 5.

$$a=2 b=3$$

Rewrite x^{2}+5x+6 as \left(x^{2}+2x\right)+\left(3x+6\right).

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

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

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

Factor out common term x+2 by using distributive property.

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

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

$$x=-2 x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Combine -6x and 3x to get -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x^{2}-7x+12 by 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Combine -3x and -28x to get -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Add -12 and 48 to get 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Subtract 30 from both sides.

$$6x^{2}-31x+36-30=5x^{2}-36x$$

Subtract 30 from 36 to get 6.

$$6x^{2}-31x+6=5x^{2}-36x$$

Subtract 5x^{2} from both sides.

$$6x^{2}-31x+6-5x^{2}=-36x$$

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

$$x^{2}-31x+6=-36x$$

Add 36x to both sides.

$$x^{2}-31x+6+36x=0$$

Combine -31x and 36x to get 5x.

$$x^{2}+5x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 5 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 6}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-5±\sqrt{25-24}}{2}$$

Add 25 to -24.

$$x=\frac{-5±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-5±1}{2}$$

Now solve the equation x=\frac{-5±1}{2} when ± is plus. Add -5 to 1.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-5±1}{2} when ± is minus. Subtract 1 from -5.

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

Divide -6 by 2.

$$x=-3$$

The equation is now solved.

$$x=-2 x=-3$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 3,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-3\right), the least common multiple of x-4,x-3,x^{2}-7x+12.

$$\left(x-3\right)\times 2x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$\left(2x-6\right)x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

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

$$2x^{2}-6x+\left(x-4\right)\times 3+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by 3.

$$2x^{2}-6x+3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Combine -6x and 3x to get -3x.

$$2x^{2}-3x-12+\left(x-4\right)\left(x-3\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x-4 by x-3 and combine like terms.

$$2x^{2}-3x-12+\left(x^{2}-7x+12\right)\times 4=30+5x^{2}-36x$$

Use the distributive property to multiply x^{2}-7x+12 by 4.

$$2x^{2}-3x-12+4x^{2}-28x+48=30+5x^{2}-36x$$

Combine 2x^{2} and 4x^{2} to get 6x^{2}.

$$6x^{2}-3x-12-28x+48=30+5x^{2}-36x$$

Combine -3x and -28x to get -31x.

$$6x^{2}-31x-12+48=30+5x^{2}-36x$$

Add -12 and 48 to get 36.

$$6x^{2}-31x+36=30+5x^{2}-36x$$

Subtract 5x^{2} from both sides.

$$6x^{2}-31x+36-5x^{2}=30-36x$$

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

$$x^{2}-31x+36=30-36x$$

Add 36x to both sides.

$$x^{2}-31x+36+36x=30$$

Combine -31x and 36x to get 5x.

$$x^{2}+5x+36=30$$

Subtract 36 from both sides.

$$x^{2}+5x=30-36$$

Subtract 36 from 30 to get -6.

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

Divide 5, the coefficient of the x term, by 2 to get \frac{5}{2}. Then add the square of \frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=-6+\frac{25}{4}$$

Add -6 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{1}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{1}{2} x+\frac{5}{2}=-\frac{1}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=-2 x=-3$$