Solve frac{-4x^{3}-7x^{2}-3x}{12x+13x^{2}-4x^{3}}=2x^3+x^2-x/2x^3-9x^2+4x

$$\frac { - 4 x ^ { 3 } - 7 x ^ { 2 } - 3 x } { 12 x + 13 x ^ { 2 } - 4 x ^ { 3 } } = \frac { 2 x ^ { 3 } + x ^ { 2 } - x } { 2 x ^ { 3 } - 9 x ^ { 2 } + 4 x }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus 4,-\frac{3}{4},0,\frac{1}{2}$$

Solution Steps

Variable x cannot be equal to any of the values -\frac{3}{4},0,\frac{1}{2},4 since division by zero is not defined. Multiply both sides of the equation by 2x\left(\frac{1}{2}x-2\right)\left(2x-1\right)\left(4x+3\right), the least common multiple of 12x+13x^{2}-4x^{3},2x^{3}-9x^{2}+4x.

$$\left(1-2x\right)\left(-4x^{3}-7x^{2}-3x\right)=\left(4x+3\right)\left(2x^{3}+x^{2}-x\right)$$

Use the distributive property to multiply 1-2x by -4x^{3}-7x^{2}-3x and combine like terms.

$$10x^{3}-x^{2}-3x+8x^{4}=\left(4x+3\right)\left(2x^{3}+x^{2}-x\right)$$

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

$$10x^{3}-x^{2}-3x+8x^{4}=8x^{4}+10x^{3}-x^{2}-3x$$

Subtract 8x^{4} from both sides.

$$10x^{3}-x^{2}-3x+8x^{4}-8x^{4}=10x^{3}-x^{2}-3x$$

Combine 8x^{4} and -8x^{4} to get 0.

$$10x^{3}-x^{2}-3x=10x^{3}-x^{2}-3x$$

Subtract 10x^{3} from both sides.

$$10x^{3}-x^{2}-3x-10x^{3}=-x^{2}-3x$$

Combine 10x^{3} and -10x^{3} to get 0.

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

Add x^{2} to both sides.

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

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

$$-3x=-3x$$

Add 3x to both sides.

$$-3x+3x=0$$

Combine -3x and 3x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Variable x cannot be equal to any of the values -\frac{3}{4},\frac{1}{2},4,0.

$$x\in \mathrm{C}\setminus -\frac{3}{4},0,\frac{1}{2},4$$

Solve for x

$$x\in \mathrm{R}\setminus 4,-\frac{3}{4},0,\frac{1}{2}$$

Solution Steps

Variable x cannot be equal to any of the values -\frac{3}{4},0,\frac{1}{2},4 since division by zero is not defined. Multiply both sides of the equation by 2x\left(\frac{1}{2}x-2\right)\left(2x-1\right)\left(4x+3\right), the least common multiple of 12x+13x^{2}-4x^{3},2x^{3}-9x^{2}+4x.

$$\left(1-2x\right)\left(-4x^{3}-7x^{2}-3x\right)=\left(4x+3\right)\left(2x^{3}+x^{2}-x\right)$$

Use the distributive property to multiply 1-2x by -4x^{3}-7x^{2}-3x and combine like terms.

$$10x^{3}-x^{2}-3x+8x^{4}=\left(4x+3\right)\left(2x^{3}+x^{2}-x\right)$$

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

$$10x^{3}-x^{2}-3x+8x^{4}=8x^{4}+10x^{3}-x^{2}-3x$$

Subtract 8x^{4} from both sides.

$$10x^{3}-x^{2}-3x+8x^{4}-8x^{4}=10x^{3}-x^{2}-3x$$

Combine 8x^{4} and -8x^{4} to get 0.

$$10x^{3}-x^{2}-3x=10x^{3}-x^{2}-3x$$

Subtract 10x^{3} from both sides.

$$10x^{3}-x^{2}-3x-10x^{3}=-x^{2}-3x$$

Combine 10x^{3} and -10x^{3} to get 0.

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

Add x^{2} to both sides.

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

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

$$-3x=-3x$$

Add 3x to both sides.

$$-3x+3x=0$$

Combine -3x and 3x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to any of the values -\frac{3}{4},\frac{1}{2},4,0.

$$x\in \mathrm{R}\setminus -\frac{3}{4},0,\frac{1}{2},4$$