Solve 2x+4/x-2-3x-6/2x+3=x^2+78/2x^2-x-6

$$\frac { 2 x + 4 } { x - 2 } - \frac { 3 x - 6 } { 2 x + 3 } = \frac { x ^ { 2 } + 78 } { 2 x ^ { 2 } - x - 6 }$$

Solve for x

$$x=3$$

Steps for Solving Linear Equation

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

$$\left(2x+3\right)\left(2x+4\right)-\left(x-2\right)\left(3x-6\right)=x^{2}+78$$

Use the distributive property to multiply 2x+3 by 2x+4 and combine like terms.

$$4x^{2}+14x+12-\left(x-2\right)\left(3x-6\right)=x^{2}+78$$

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

$$4x^{2}+14x+12-\left(3x^{2}-12x+12\right)=x^{2}+78$$

To find the opposite of 3x^{2}-12x+12, find the opposite of each term.

$$4x^{2}+14x+12-3x^{2}+12x-12=x^{2}+78$$

Combine 4x^{2} and -3x^{2} to get x^{2}.

$$x^{2}+14x+12+12x-12=x^{2}+78$$

Combine 14x and 12x to get 26x.

$$x^{2}+26x+12-12=x^{2}+78$$

Subtract 12 from 12 to get 0.

$$x^{2}+26x=x^{2}+78$$

Subtract x^{2} from both sides.

$$x^{2}+26x-x^{2}=78$$

Combine x^{2} and -x^{2} to get 0.

$$26x=78$$

Divide both sides by 26.

$$x=\frac{78}{26}$$

Divide 78 by 26 to get 3.

$$x=3$$