Solve 2x^2+x-3/2x-7+2x^2-5x+2/4x-6

$$\frac{ 2 { x }^{ 2 } +x-3 }{ 2x-7 } + \frac{ 2 { x }^{ 2 } -5x+2 }{ 4x-6 }$$

Evaluate

$$\frac{12x^{3}-32x^{2}+21x+4}{2\left(2x-7\right)\left(2x-3\right)}$$

Short Solution Steps

Factor 4x-6.

$$\frac{2x^{2}+x-3}{2x-7}+\frac{2x^{2}-5x+2}{2\left(2x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x-7 and 2\left(2x-3\right) is 2\left(2x-7\right)\left(2x-3\right). Multiply \frac{2x^{2}+x-3}{2x-7} times \frac{2\left(2x-3\right)}{2\left(2x-3\right)}. Multiply \frac{2x^{2}-5x+2}{2\left(2x-3\right)} times \frac{2x-7}{2x-7}.

$$\frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)}{2\left(2x-7\right)\left(2x-3\right)}+\frac{\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)}$$

Since \frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)}{2\left(2x-7\right)\left(2x-3\right)} and \frac{\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)+\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)}$$

Do the multiplications in \left(2x^{2}+x-3\right)\times 2\left(2x-3\right)+\left(2x^{2}-5x+2\right)\left(2x-7\right).

$$\frac{8x^{3}-12x^{2}+4x^{2}-6x-12x+18+4x^{3}-14x^{2}-10x^{2}+35x+4x-14}{2\left(2x-7\right)\left(2x-3\right)}$$

Combine like terms in 8x^{3}-12x^{2}+4x^{2}-6x-12x+18+4x^{3}-14x^{2}-10x^{2}+35x+4x-14.

$$\frac{12x^{3}-32x^{2}+21x+4}{2\left(2x-7\right)\left(2x-3\right)}$$

Expand 2\left(2x-7\right)\left(2x-3\right).

$$\frac{12x^{3}-32x^{2}+21x+4}{8x^{2}-40x+42}$$

Expand

$$\frac{12x^{3}-32x^{2}+21x+4}{2\left(2x-7\right)\left(2x-3\right)}$$

Short Solution Steps

Factor 4x-6.

$$\frac{2x^{2}+x-3}{2x-7}+\frac{2x^{2}-5x+2}{2\left(2x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2x-7 and 2\left(2x-3\right) is 2\left(2x-7\right)\left(2x-3\right). Multiply \frac{2x^{2}+x-3}{2x-7} times \frac{2\left(2x-3\right)}{2\left(2x-3\right)}. Multiply \frac{2x^{2}-5x+2}{2\left(2x-3\right)} times \frac{2x-7}{2x-7}.

$$\frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)}{2\left(2x-7\right)\left(2x-3\right)}+\frac{\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)}$$

Since \frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)}{2\left(2x-7\right)\left(2x-3\right)} and \frac{\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}+x-3\right)\times 2\left(2x-3\right)+\left(2x^{2}-5x+2\right)\left(2x-7\right)}{2\left(2x-7\right)\left(2x-3\right)}$$

Do the multiplications in \left(2x^{2}+x-3\right)\times 2\left(2x-3\right)+\left(2x^{2}-5x+2\right)\left(2x-7\right).

$$\frac{8x^{3}-12x^{2}+4x^{2}-6x-12x+18+4x^{3}-14x^{2}-10x^{2}+35x+4x-14}{2\left(2x-7\right)\left(2x-3\right)}$$

Combine like terms in 8x^{3}-12x^{2}+4x^{2}-6x-12x+18+4x^{3}-14x^{2}-10x^{2}+35x+4x-14.

$$\frac{12x^{3}-32x^{2}+21x+4}{2\left(2x-7\right)\left(2x-3\right)}$$

Expand 2\left(2x-7\right)\left(2x-3\right).

$$\frac{12x^{3}-32x^{2}+21x+4}{8x^{2}-40x+42}$$