Solve 26*x-1/x-2-x-3/x-4*x-4/x-3-x-2/x-1

$$26 \cdot \frac { x - 1 } { x - 2 } - \frac { x - 3 } { x - 4 } \cdot \frac { x - 4 } { x - 3 } - \frac { x - 2 } { x - 1 }$$

Evaluate

$$\frac{24x^{2}-45x+20}{\left(x-2\right)\left(x-1\right)}$$

Short Solution Steps

Express 26\times \frac{x-1}{x-2} as a single fraction.

$$\frac{26\left(x-1\right)}{x-2}-\frac{x-3}{x-4}\times \frac{x-4}{x-3}-\frac{x-2}{x-1}$$

Multiply \frac{x-3}{x-4} times \frac{x-4}{x-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{26\left(x-1\right)}{x-2}-\frac{\left(x-3\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x-1}$$

Cancel out \left(x-4\right)\left(x-3\right) in both numerator and denominator.

$$\frac{26\left(x-1\right)}{x-2}-1-\frac{x-2}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

$$\frac{26\left(x-1\right)}{x-2}-\frac{x-2}{x-2}-\frac{x-2}{x-1}$$

Since \frac{26\left(x-1\right)}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{26\left(x-1\right)-\left(x-2\right)}{x-2}-\frac{x-2}{x-1}$$

Do the multiplications in 26\left(x-1\right)-\left(x-2\right).

$$\frac{26x-26-x+2}{x-2}-\frac{x-2}{x-1}$$

Combine like terms in 26x-26-x+2.

$$\frac{25x-24}{x-2}-\frac{x-2}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x-1 is \left(x-2\right)\left(x-1\right). Multiply \frac{25x-24}{x-2} times \frac{x-1}{x-1}. Multiply \frac{x-2}{x-1} times \frac{x-2}{x-2}.

$$\frac{\left(25x-24\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}-\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(25x-24\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)} and \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(25x-24\right)\left(x-1\right)-\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(25x-24\right)\left(x-1\right)-\left(x-2\right)\left(x-2\right).

$$\frac{25x^{2}-25x-24x+24-x^{2}+2x+2x-4}{\left(x-2\right)\left(x-1\right)}$$

Combine like terms in 25x^{2}-25x-24x+24-x^{2}+2x+2x-4.

$$\frac{24x^{2}-45x+20}{\left(x-2\right)\left(x-1\right)}$$

Expand \left(x-2\right)\left(x-1\right).

$$\frac{24x^{2}-45x+20}{x^{2}-3x+2}$$

Expand

$$\frac{24x^{2}-45x+20}{\left(x-2\right)\left(x-1\right)}$$

Short Solution Steps

Express 26\times \frac{x-1}{x-2} as a single fraction.

$$\frac{26\left(x-1\right)}{x-2}-\frac{x-3}{x-4}\times \frac{x-4}{x-3}-\frac{x-2}{x-1}$$

Multiply \frac{x-3}{x-4} times \frac{x-4}{x-3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{26\left(x-1\right)}{x-2}-\frac{\left(x-3\right)\left(x-4\right)}{\left(x-4\right)\left(x-3\right)}-\frac{x-2}{x-1}$$

Cancel out \left(x-4\right)\left(x-3\right) in both numerator and denominator.

$$\frac{26\left(x-1\right)}{x-2}-1-\frac{x-2}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{x-2}{x-2}.

$$\frac{26\left(x-1\right)}{x-2}-\frac{x-2}{x-2}-\frac{x-2}{x-1}$$

Since \frac{26\left(x-1\right)}{x-2} and \frac{x-2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{26\left(x-1\right)-\left(x-2\right)}{x-2}-\frac{x-2}{x-1}$$

Do the multiplications in 26\left(x-1\right)-\left(x-2\right).

$$\frac{26x-26-x+2}{x-2}-\frac{x-2}{x-1}$$

Combine like terms in 26x-26-x+2.

$$\frac{25x-24}{x-2}-\frac{x-2}{x-1}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-2 and x-1 is \left(x-2\right)\left(x-1\right). Multiply \frac{25x-24}{x-2} times \frac{x-1}{x-1}. Multiply \frac{x-2}{x-1} times \frac{x-2}{x-2}.

$$\frac{\left(25x-24\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)}-\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)}$$

Since \frac{\left(25x-24\right)\left(x-1\right)}{\left(x-2\right)\left(x-1\right)} and \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(25x-24\right)\left(x-1\right)-\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x-1\right)}$$

Do the multiplications in \left(25x-24\right)\left(x-1\right)-\left(x-2\right)\left(x-2\right).

$$\frac{25x^{2}-25x-24x+24-x^{2}+2x+2x-4}{\left(x-2\right)\left(x-1\right)}$$

Combine like terms in 25x^{2}-25x-24x+24-x^{2}+2x+2x-4.

$$\frac{24x^{2}-45x+20}{\left(x-2\right)\left(x-1\right)}$$

Expand \left(x-2\right)\left(x-1\right).

$$\frac{24x^{2}-45x+20}{x^{2}-3x+2}$$