Solve 2/2x-6-4x/x-4

$$\frac { 2 } { 2 x - 6 } - \frac { 4 x } { x - 4 }$$

Evaluate

$$\frac{-4x^{2}+13x-4}{\left(x-4\right)\left(x-3\right)}$$

Short Solution Steps

Factor 2x-6.

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

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

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

Since \frac{2\left(x-4\right)}{2\left(x-4\right)\left(x-3\right)} and \frac{4x\times 2\left(x-3\right)}{2\left(x-4\right)\left(x-3\right)} have the same denominator, subtract them by subtracting their numerators.

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

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

$$\frac{2x-8-8x^{2}+24x}{2\left(x-4\right)\left(x-3\right)}$$

Combine like terms in 2x-8-8x^{2}+24x.

$$\frac{26x-8-8x^{2}}{2\left(x-4\right)\left(x-3\right)}$$

Factor the expressions that are not already factored in \frac{26x-8-8x^{2}}{2\left(x-4\right)\left(x-3\right)}.

$$\frac{-2\times 4\left(x-\left(-\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)}{2\left(x-4\right)\left(x-3\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{-4\left(x-\left(-\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)}{\left(x-4\right)\left(x-3\right)}$$

Expand \left(x-4\right)\left(x-3\right).

$$\frac{-4\left(x-\left(-\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)\left(x-\left(\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)}{x^{2}-7x+12}$$

To find the opposite of -\frac{1}{8}\sqrt{105}+\frac{13}{8}, find the opposite of each term.

$$\frac{-4\left(x-\left(-\frac{1}{8}\sqrt{105}\right)-\frac{13}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)}{x^{2}-7x+12}$$

The opposite of -\frac{1}{8}\sqrt{105} is \frac{1}{8}\sqrt{105}.

$$\frac{-4\left(x+\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)\left(x-\left(\frac{1}{8}\sqrt{105}+\frac{13}{8}\right)\right)}{x^{2}-7x+12}$$

To find the opposite of \frac{1}{8}\sqrt{105}+\frac{13}{8}, find the opposite of each term.

$$\frac{-4\left(x+\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Use the distributive property to multiply -4 by x+\frac{1}{8}\sqrt{105}-\frac{13}{8}.

$$\frac{\left(-4x-4\times \frac{1}{8}\sqrt{105}-4\left(-\frac{13}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -4 and \frac{1}{8} to get \frac{-4}{8}.

$$\frac{\left(-4x+\frac{-4}{8}\sqrt{105}-4\left(-\frac{13}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Reduce the fraction \frac{-4}{8} to lowest terms by extracting and canceling out 4.

$$\frac{\left(-4x-\frac{1}{2}\sqrt{105}-4\left(-\frac{13}{8}\right)\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Express -4\left(-\frac{13}{8}\right) as a single fraction.

$$\frac{\left(-4x-\frac{1}{2}\sqrt{105}+\frac{-4\left(-13\right)}{8}\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -4 and -13 to get 52.

$$\frac{\left(-4x-\frac{1}{2}\sqrt{105}+\frac{52}{8}\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Reduce the fraction \frac{52}{8} to lowest terms by extracting and canceling out 4.

$$\frac{\left(-4x-\frac{1}{2}\sqrt{105}+\frac{13}{2}\right)\left(x-\frac{1}{8}\sqrt{105}-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Apply the distributive property by multiplying each term of -4x-\frac{1}{2}\sqrt{105}+\frac{13}{2} by each term of x-\frac{1}{8}\sqrt{105}-\frac{13}{8}.

$$\frac{-4x^{2}-4x\left(-\frac{1}{8}\right)\sqrt{105}-4x\left(-\frac{13}{8}\right)-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\sqrt{105}\left(-\frac{1}{8}\right)\sqrt{105}-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply \sqrt{105} and \sqrt{105} to get 105.

$$\frac{-4x^{2}-4x\left(-\frac{1}{8}\right)\sqrt{105}-4x\left(-\frac{13}{8}\right)-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Express -4\left(-\frac{1}{8}\right) as a single fraction.

$$\frac{-4x^{2}+\frac{-4\left(-1\right)}{8}x\sqrt{105}-4x\left(-\frac{13}{8}\right)-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -4 and -1 to get 4.

$$\frac{-4x^{2}+\frac{4}{8}x\sqrt{105}-4x\left(-\frac{13}{8}\right)-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$\frac{-4x^{2}+\frac{1}{2}x\sqrt{105}-4x\left(-\frac{13}{8}\right)-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Express -4\left(-\frac{13}{8}\right) as a single fraction.

$$\frac{-4x^{2}+\frac{1}{2}x\sqrt{105}+\frac{-4\left(-13\right)}{8}x-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -4 and -13 to get 52.

$$\frac{-4x^{2}+\frac{1}{2}x\sqrt{105}+\frac{52}{8}x-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Reduce the fraction \frac{52}{8} to lowest terms by extracting and canceling out 4.

$$\frac{-4x^{2}+\frac{1}{2}x\sqrt{105}+\frac{13}{2}x-\frac{1}{2}\sqrt{105}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Combine \frac{1}{2}x\sqrt{105} and -\frac{1}{2}\sqrt{105}x to get 0.

$$\frac{-4x^{2}+\frac{13}{2}x-\frac{1}{2}\times 105\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Express -\frac{1}{2}\times 105 as a single fraction.

$$\frac{-4x^{2}+\frac{13}{2}x+\frac{-105}{2}\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Fraction \frac{-105}{2} can be rewritten as -\frac{105}{2} by extracting the negative sign.

$$\frac{-4x^{2}+\frac{13}{2}x-\frac{105}{2}\left(-\frac{1}{8}\right)-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -\frac{105}{2} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-4x^{2}+\frac{13}{2}x+\frac{-105\left(-1\right)}{2\times 8}-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Do the multiplications in the fraction \frac{-105\left(-1\right)}{2\times 8}.

$$\frac{-4x^{2}+\frac{13}{2}x+\frac{105}{16}-\frac{1}{2}\sqrt{105}\left(-\frac{13}{8}\right)+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply -\frac{1}{2} times -\frac{13}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-4x^{2}+\frac{13}{2}x+\frac{105}{16}+\frac{-\left(-13\right)}{2\times 8}\sqrt{105}+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Do the multiplications in the fraction \frac{-\left(-13\right)}{2\times 8}.

$$\frac{-4x^{2}+\frac{13}{2}x+\frac{105}{16}+\frac{13}{16}\sqrt{105}+\frac{13}{2}x+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Combine \frac{13}{2}x and \frac{13}{2}x to get 13x.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13}{16}\sqrt{105}+\frac{13}{2}\left(-\frac{1}{8}\right)\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply \frac{13}{2} times -\frac{1}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13}{16}\sqrt{105}+\frac{13\left(-1\right)}{2\times 8}\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Do the multiplications in the fraction \frac{13\left(-1\right)}{2\times 8}.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13}{16}\sqrt{105}+\frac{-13}{16}\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Fraction \frac{-13}{16} can be rewritten as -\frac{13}{16} by extracting the negative sign.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13}{16}\sqrt{105}-\frac{13}{16}\sqrt{105}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Combine \frac{13}{16}\sqrt{105} and -\frac{13}{16}\sqrt{105} to get 0.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13}{2}\left(-\frac{13}{8}\right)}{x^{2}-7x+12}$$

Multiply \frac{13}{2} times -\frac{13}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{13\left(-13\right)}{2\times 8}}{x^{2}-7x+12}$$

Do the multiplications in the fraction \frac{13\left(-13\right)}{2\times 8}.

$$\frac{-4x^{2}+13x+\frac{105}{16}+\frac{-169}{16}}{x^{2}-7x+12}$$

Fraction \frac{-169}{16} can be rewritten as -\frac{169}{16} by extracting the negative sign.

$$\frac{-4x^{2}+13x+\frac{105}{16}-\frac{169}{16}}{x^{2}-7x+12}$$

Since \frac{105}{16} and \frac{169}{16} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-4x^{2}+13x+\frac{105-169}{16}}{x^{2}-7x+12}$$

Subtract 169 from 105 to get -64.

$$\frac{-4x^{2}+13x+\frac{-64}{16}}{x^{2}-7x+12}$$

Divide -64 by 16 to get -4.

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

Differentiate w.r.t. x

$$\frac{\left(3x-8\right)\left(5x-16\right)}{\left(\left(x-4\right)\left(x-3\right)\right)^{2}}$$