Solve k^2-4/k^2+4-1-4k^2/4k^2+1

$$\frac { k ^ { 2 } - 4 } { k ^ { 2 } + 4 } - \frac { 1 - 4 k ^ { 2 } } { 4 k ^ { 2 } + 1 }$$

Evaluate

$$\frac{8\left(k^{4}-1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k^{2}+4 and 4k^{2}+1 is \left(4k^{2}+1\right)\left(k^{2}+4\right). Multiply \frac{k^{2}-4}{k^{2}+4} times \frac{4k^{2}+1}{4k^{2}+1}. Multiply \frac{1-4k^{2}}{4k^{2}+1} times \frac{k^{2}+4}{k^{2}+4}.

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

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

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

Do the multiplications in \left(k^{2}-4\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right).

$$\frac{4k^{4}+k^{2}-16k^{2}-4-k^{2}-4+4k^{4}+16k^{2}}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Combine like terms in 4k^{4}+k^{2}-16k^{2}-4-k^{2}-4+4k^{4}+16k^{2}.

$$\frac{8k^{4}-8}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Expand \left(4k^{2}+1\right)\left(k^{2}+4\right).

$$\frac{8k^{4}-8}{4k^{4}+17k^{2}+4}$$

Expand

$$\frac{8\left(k^{4}-1\right)}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of k^{2}+4 and 4k^{2}+1 is \left(4k^{2}+1\right)\left(k^{2}+4\right). Multiply \frac{k^{2}-4}{k^{2}+4} times \frac{4k^{2}+1}{4k^{2}+1}. Multiply \frac{1-4k^{2}}{4k^{2}+1} times \frac{k^{2}+4}{k^{2}+4}.

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

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

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

Do the multiplications in \left(k^{2}-4\right)\left(4k^{2}+1\right)-\left(1-4k^{2}\right)\left(k^{2}+4\right).

$$\frac{4k^{4}+k^{2}-16k^{2}-4-k^{2}-4+4k^{4}+16k^{2}}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Combine like terms in 4k^{4}+k^{2}-16k^{2}-4-k^{2}-4+4k^{4}+16k^{2}.

$$\frac{8k^{4}-8}{\left(4k^{2}+1\right)\left(k^{2}+4\right)}$$

Expand \left(4k^{2}+1\right)\left(k^{2}+4\right).

$$\frac{8k^{4}-8}{4k^{4}+17k^{2}+4}$$