Solve 4x+11/2x-6+3x/x^2-11x+24=

$$\frac { 4 x + 11 } { 2 x - 6 } + \frac { 3 x } { x ^ { 2 } - 11 x + 24 } =$$

Evaluate

$$\frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}$$

Short Solution Steps

Factor 2x-6. Factor x^{2}-11x+24.

$$\frac{4x+11}{2\left(x-3\right)}+\frac{3x}{\left(x-8\right)\left(x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-3\right) and \left(x-8\right)\left(x-3\right) is 2\left(x-8\right)\left(x-3\right). Multiply \frac{4x+11}{2\left(x-3\right)} times \frac{x-8}{x-8}. Multiply \frac{3x}{\left(x-8\right)\left(x-3\right)} times \frac{2}{2}.

$$\frac{\left(4x+11\right)\left(x-8\right)}{2\left(x-8\right)\left(x-3\right)}+\frac{2\times 3x}{2\left(x-8\right)\left(x-3\right)}$$

Since \frac{\left(4x+11\right)\left(x-8\right)}{2\left(x-8\right)\left(x-3\right)} and \frac{2\times 3x}{2\left(x-8\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x+11\right)\left(x-8\right)+2\times 3x}{2\left(x-8\right)\left(x-3\right)}$$

Do the multiplications in \left(4x+11\right)\left(x-8\right)+2\times 3x.

$$\frac{4x^{2}-32x+11x-88+6x}{2\left(x-8\right)\left(x-3\right)}$$

Combine like terms in 4x^{2}-32x+11x-88+6x.

$$\frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}.

$$\frac{4\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{2\left(x-8\right)\left(x-3\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{\left(x-8\right)\left(x-3\right)}$$

Expand \left(x-8\right)\left(x-3\right).

$$\frac{2\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{x^{2}-11x+24}$$

To find the opposite of -\frac{1}{8}\sqrt{1633}+\frac{15}{8}, find the opposite of each term.

$$\frac{2\left(x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{x^{2}-11x+24}$$

To find the opposite of \frac{1}{8}\sqrt{1633}+\frac{15}{8}, find the opposite of each term.

$$\frac{2\left(x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)\left(x-\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)}{x^{2}-11x+24}$$

Use the distributive property to multiply 2 by x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}.

$$\frac{\left(2x+\frac{1}{4}\sqrt{1633}-\frac{15}{4}\right)\left(x-\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)}{x^{2}-11x+24}$$

Use the distributive property to multiply 2x+\frac{1}{4}\sqrt{1633}-\frac{15}{4} by x-\frac{1}{8}\sqrt{1633}-\frac{15}{8} and combine like terms.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1}{32}\left(\sqrt{1633}\right)^{2}+\frac{225}{32}}{x^{2}-11x+24}$$

The square of \sqrt{1633} is 1633.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1}{32}\times 1633+\frac{225}{32}}{x^{2}-11x+24}$$

Multiply -\frac{1}{32} and 1633 to get -\frac{1633}{32}.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1633}{32}+\frac{225}{32}}{x^{2}-11x+24}$$

Add -\frac{1633}{32} and \frac{225}{32} to get -44.

$$\frac{2x^{2}-\frac{15}{2}x-44}{x^{2}-11x+24}$$

Expand

$$\frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}$$

Short Solution Steps

Factor 2x-6. Factor x^{2}-11x+24.

$$\frac{4x+11}{2\left(x-3\right)}+\frac{3x}{\left(x-8\right)\left(x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-3\right) and \left(x-8\right)\left(x-3\right) is 2\left(x-8\right)\left(x-3\right). Multiply \frac{4x+11}{2\left(x-3\right)} times \frac{x-8}{x-8}. Multiply \frac{3x}{\left(x-8\right)\left(x-3\right)} times \frac{2}{2}.

$$\frac{\left(4x+11\right)\left(x-8\right)}{2\left(x-8\right)\left(x-3\right)}+\frac{2\times 3x}{2\left(x-8\right)\left(x-3\right)}$$

Since \frac{\left(4x+11\right)\left(x-8\right)}{2\left(x-8\right)\left(x-3\right)} and \frac{2\times 3x}{2\left(x-8\right)\left(x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x+11\right)\left(x-8\right)+2\times 3x}{2\left(x-8\right)\left(x-3\right)}$$

Do the multiplications in \left(4x+11\right)\left(x-8\right)+2\times 3x.

$$\frac{4x^{2}-32x+11x-88+6x}{2\left(x-8\right)\left(x-3\right)}$$

Combine like terms in 4x^{2}-32x+11x-88+6x.

$$\frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{4x^{2}-15x-88}{2\left(x-8\right)\left(x-3\right)}.

$$\frac{4\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{2\left(x-8\right)\left(x-3\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{\left(x-8\right)\left(x-3\right)}$$

Expand \left(x-8\right)\left(x-3\right).

$$\frac{2\left(x-\left(-\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{x^{2}-11x+24}$$

To find the opposite of -\frac{1}{8}\sqrt{1633}+\frac{15}{8}, find the opposite of each term.

$$\frac{2\left(x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{1633}+\frac{15}{8}\right)\right)}{x^{2}-11x+24}$$

To find the opposite of \frac{1}{8}\sqrt{1633}+\frac{15}{8}, find the opposite of each term.

$$\frac{2\left(x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)\left(x-\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)}{x^{2}-11x+24}$$

Use the distributive property to multiply 2 by x+\frac{1}{8}\sqrt{1633}-\frac{15}{8}.

$$\frac{\left(2x+\frac{1}{4}\sqrt{1633}-\frac{15}{4}\right)\left(x-\frac{1}{8}\sqrt{1633}-\frac{15}{8}\right)}{x^{2}-11x+24}$$

Use the distributive property to multiply 2x+\frac{1}{4}\sqrt{1633}-\frac{15}{4} by x-\frac{1}{8}\sqrt{1633}-\frac{15}{8} and combine like terms.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1}{32}\left(\sqrt{1633}\right)^{2}+\frac{225}{32}}{x^{2}-11x+24}$$

The square of \sqrt{1633} is 1633.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1}{32}\times 1633+\frac{225}{32}}{x^{2}-11x+24}$$

Multiply -\frac{1}{32} and 1633 to get -\frac{1633}{32}.

$$\frac{2x^{2}-\frac{15}{2}x-\frac{1633}{32}+\frac{225}{32}}{x^{2}-11x+24}$$

Add -\frac{1633}{32} and \frac{225}{32} to get -44.

$$\frac{2x^{2}-\frac{15}{2}x-44}{x^{2}-11x+24}$$