Solve 6/-3-3/a-1-a+1/a^2-4a+3

$$\frac { 6 } { - 3 } - \frac { 3 } { a - 1 } - \frac { a + 1 } { a ^ { 2 } - 4 a + 3 }$$

Evaluate

$$\frac{2\left(1+2a-a^{2}\right)}{\left(a-3\right)\left(a-1\right)}$$

Short Solution Steps

Divide 6 by -3 to get -2.

$$-2-\frac{3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

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

$$-\frac{2\left(a-1\right)}{a-1}-\frac{3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Since -\frac{2\left(a-1\right)}{a-1} and \frac{3}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2\left(a-1\right)-3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Do the multiplications in -2\left(a-1\right)-3.

$$\frac{-2a+2-3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Combine like terms in -2a+2-3.

$$\frac{-2a-1}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Factor a^{2}-4a+3.

$$\frac{-2a-1}{a-1}-\frac{a+1}{\left(a-3\right)\left(a-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and \left(a-3\right)\left(a-1\right) is \left(a-3\right)\left(a-1\right). Multiply \frac{-2a-1}{a-1} times \frac{a-3}{a-3}.

$$\frac{\left(-2a-1\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)}-\frac{a+1}{\left(a-3\right)\left(a-1\right)}$$

Since \frac{\left(-2a-1\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)} and \frac{a+1}{\left(a-3\right)\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-2a-1\right)\left(a-3\right)-\left(a+1\right)}{\left(a-3\right)\left(a-1\right)}$$

Do the multiplications in \left(-2a-1\right)\left(a-3\right)-\left(a+1\right).

$$\frac{-2a^{2}+6a-a+3-a-1}{\left(a-3\right)\left(a-1\right)}$$

Combine like terms in -2a^{2}+6a-a+3-a-1.

$$\frac{-2a^{2}+4a+2}{\left(a-3\right)\left(a-1\right)}$$

Expand \left(a-3\right)\left(a-1\right).

$$\frac{-2a^{2}+4a+2}{a^{2}-4a+3}$$

Expand

$$-\frac{2\left(a^{2}-2a-1\right)}{\left(a-3\right)\left(a-1\right)}$$

Short Solution Steps

Divide 6 by -3 to get -2.

$$-2-\frac{3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

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

$$-\frac{2\left(a-1\right)}{a-1}-\frac{3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Since -\frac{2\left(a-1\right)}{a-1} and \frac{3}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2\left(a-1\right)-3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Do the multiplications in -2\left(a-1\right)-3.

$$\frac{-2a+2-3}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Combine like terms in -2a+2-3.

$$\frac{-2a-1}{a-1}-\frac{a+1}{a^{2}-4a+3}$$

Factor a^{2}-4a+3.

$$\frac{-2a-1}{a-1}-\frac{a+1}{\left(a-3\right)\left(a-1\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-1 and \left(a-3\right)\left(a-1\right) is \left(a-3\right)\left(a-1\right). Multiply \frac{-2a-1}{a-1} times \frac{a-3}{a-3}.

$$\frac{\left(-2a-1\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)}-\frac{a+1}{\left(a-3\right)\left(a-1\right)}$$

Since \frac{\left(-2a-1\right)\left(a-3\right)}{\left(a-3\right)\left(a-1\right)} and \frac{a+1}{\left(a-3\right)\left(a-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(-2a-1\right)\left(a-3\right)-\left(a+1\right)}{\left(a-3\right)\left(a-1\right)}$$

Do the multiplications in \left(-2a-1\right)\left(a-3\right)-\left(a+1\right).

$$\frac{-2a^{2}+6a-a+3-a-1}{\left(a-3\right)\left(a-1\right)}$$

Combine like terms in -2a^{2}+6a-a+3-a-1.

$$\frac{-2a^{2}+4a+2}{\left(a-3\right)\left(a-1\right)}$$

Expand \left(a-3\right)\left(a-1\right).

$$\frac{-2a^{2}+4a+2}{a^{2}-4a+3}$$