Solve 3x+54/6x^{2+3x}+4x^2+9/4x^2-1=x+3/2/3x-frac{8}{3}*frac{x^2}{(1-4x^2)}

$$\frac { 3 x + 54 } { 6 x ^ { 2 } + 3 x } + \frac { 4 x ^ { 2 } + 9 } { 4 x ^ { 2 } - 1 } = \frac { x + \frac { 3 } { 2 } } { 3 x } - \frac { 8 } { 3 } \cdot \frac { x ^ { 2 } } { ( 1 - 4 x ^ { 2 } ) }$$

Solve for x

$$x=\frac{15}{38}\approx 0.394736842$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -\frac{1}{2},0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by 3x\left(2x-1\right)\left(2x+1\right), the least common multiple of 6x^{2}+3x,4x^{2}-1,3x,3,1-4x^{2}.

$$\left(2x-1\right)\left(3x+54\right)+3x\left(4x^{2}+9\right)=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Use the distributive property to multiply 2x-1 by 3x+54 and combine like terms.

$$6x^{2}+105x-54+3x\left(4x^{2}+9\right)=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Use the distributive property to multiply 3x by 4x^{2}+9.

$$6x^{2}+105x-54+12x^{3}+27x=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

Combine 105x and 27x to get 132x.

$$6x^{2}+132x-54+12x^{3}=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)xx^{2}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$6x^{2}+132x-54+12x^{3}=\left(4x^{2}-1\right)\left(x+\frac{3}{2}\right)-\frac{8}{3}\left(-3\right)x^{3}$$

Use the distributive property to multiply 4x^{2}-1 by x+\frac{3}{2}.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}-\frac{8}{3}\left(-3\right)x^{3}$$

Multiply \frac{8}{3} and -3 to get -8.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}-\left(-8x^{3}\right)$$

The opposite of -8x^{3} is 8x^{3}.

$$6x^{2}+132x-54+12x^{3}=4x^{3}+6x^{2}-x-\frac{3}{2}+8x^{3}$$

Combine 4x^{3} and 8x^{3} to get 12x^{3}.

$$6x^{2}+132x-54+12x^{3}=12x^{3}+6x^{2}-x-\frac{3}{2}$$

Subtract 12x^{3} from both sides.

$$6x^{2}+132x-54+12x^{3}-12x^{3}=6x^{2}-x-\frac{3}{2}$$

Combine 12x^{3} and -12x^{3} to get 0.

$$6x^{2}+132x-54=6x^{2}-x-\frac{3}{2}$$

Subtract 6x^{2} from both sides.

$$6x^{2}+132x-54-6x^{2}=-x-\frac{3}{2}$$

Combine 6x^{2} and -6x^{2} to get 0.

$$132x-54=-x-\frac{3}{2}$$

Add x to both sides.

$$132x-54+x=-\frac{3}{2}$$

Combine 132x and x to get 133x.

$$133x-54=-\frac{3}{2}$$

Add 54 to both sides.

$$133x=-\frac{3}{2}+54$$

Add -\frac{3}{2} and 54 to get \frac{105}{2}.

$$133x=\frac{105}{2}$$

Divide both sides by 133.

$$x=\frac{\frac{105}{2}}{133}$$

Express \frac{\frac{105}{2}}{133} as a single fraction.

$$x=\frac{105}{2\times 133}$$

Multiply 2 and 133 to get 266.

$$x=\frac{105}{266}$$

Reduce the fraction \frac{105}{266} to lowest terms by extracting and canceling out 7.

$$x=\frac{15}{38}$$