Solve 2a-1/2a-2a/2a-1-1/2a-4a^2

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

Evaluate

$$-\frac{1}{a}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in 4a^{2}-2a-2a+1-4a^{2}.

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

Factor 2a-4a^{2}.

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

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

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

Since \frac{-4a+1}{2a\left(2a-1\right)} and \frac{-1}{2a\left(2a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -4a+1-\left(-1\right).

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

Combine like terms in -4a+1+1.

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

Factor the expressions that are not already factored in \frac{-4a+2}{2a\left(2a-1\right)}.

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

Extract the negative sign in 1-2a.

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

Cancel out 2\left(2a-1\right) in both numerator and denominator.

$$\frac{-1}{a}$$

Expand

$$-\frac{1}{a}$$

Short Solution Steps

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

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

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

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

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

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

Combine like terms in 4a^{2}-2a-2a+1-4a^{2}.

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

Factor 2a-4a^{2}.

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

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

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

Since \frac{-4a+1}{2a\left(2a-1\right)} and \frac{-1}{2a\left(2a-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -4a+1-\left(-1\right).

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

Combine like terms in -4a+1+1.

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

Factor the expressions that are not already factored in \frac{-4a+2}{2a\left(2a-1\right)}.

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

Extract the negative sign in 1-2a.

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

Cancel out 2\left(2a-1\right) in both numerator and denominator.

$$\frac{-1}{a}$$