Solve (a^2+1/2ab)+(-1/4ab+1/2b^2)+(-1/4ab-frac{1}{5}b^2)

$$( a ^ { 2 } + \frac { 1 } { 2 } a b ) + ( - \frac { 1 } { 4 } a b + \frac { 1 } { 2 } b ^ { 2 } ) + ( - \frac { 1 } { 4 } a b - \frac { 1 } { 5 } b ^ { 2 } )$$

Evaluate

$$\frac{3b^{2}}{10}+a^{2}$$

Solution Steps

Combine \frac{1}{2}ab and -\frac{1}{4}ab to get \frac{1}{4}ab.

$$a^{2}+\frac{1}{4}ab+\frac{1}{2}b^{2}-\frac{1}{4}ab-\frac{1}{5}b^{2}$$

Combine \frac{1}{4}ab and -\frac{1}{4}ab to get 0.

$$a^{2}+\frac{1}{2}b^{2}-\frac{1}{5}b^{2}$$

Combine \frac{1}{2}b^{2} and -\frac{1}{5}b^{2} to get \frac{3}{10}b^{2}.

$$a^{2}+\frac{3}{10}b^{2}$$

Factor

$$\frac{10a^{2}+3b^{2}}{10}$$

Solution Steps

Factor out \frac{1}{20}.

$$\frac{20a^{2}+10ab-5ab+10b^{2}-5ab-4b^{2}}{20}$$

Consider 20a^{2}+10ab-5ab+10b^{2}-5ab-4b^{2}. Multiply and combine like terms.

$$20a^{2}+6b^{2}$$

Consider 20a^{2}+6b^{2}. Factor out 2.

$$2\left(10a^{2}+3b^{2}\right)$$

Rewrite the complete factored expression.

$$\frac{10a^{2}+3b^{2}}{10}$$