Solve a^2/25+b^2/16-ab/10

$$\frac { a ^ { 2 } } { 25 } + \frac { b ^ { 2 } } { 16 } - \frac { a b } { 10 }$$

Evaluate

$$\frac{\left(4a-5b\right)^{2}}{400}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 25 and 16 is 400. Multiply \frac{a^{2}}{25} times \frac{16}{16}. Multiply \frac{b^{2}}{16} times \frac{25}{25}.

$$\frac{16a^{2}}{400}+\frac{25b^{2}}{400}-\frac{ab}{10}$$

Since \frac{16a^{2}}{400} and \frac{25b^{2}}{400} have the same denominator, add them by adding their numerators.

$$\frac{16a^{2}+25b^{2}}{400}-\frac{ab}{10}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 400 and 10 is 400. Multiply \frac{ab}{10} times \frac{40}{40}.

$$\frac{16a^{2}+25b^{2}}{400}-\frac{40ab}{400}$$

Since \frac{16a^{2}+25b^{2}}{400} and \frac{40ab}{400} have the same denominator, subtract them by subtracting their numerators.

$$\frac{16a^{2}+25b^{2}-40ab}{400}$$

Factor

$$\frac{\left(4a-5b\right)^{2}}{400}$$

Solution Steps

Factor out \frac{1}{400}.

$$\frac{16a^{2}+25b^{2}-40ab}{400}$$

Consider 16a^{2}+25b^{2}-40ab. Use the perfect square formula, p^{2}-2pq+q^{2}=\left(p-q\right)^{2}, where p=4a and q=5b.

$$\left(4a-5b\right)^{2}$$

Rewrite the complete factored expression.

$$\frac{\left(4a-5b\right)^{2}}{400}$$