Solve 1/7a^2-1/7ab+3/49(a-b)^2-6/49a^2+frac{6}{49}ab

$$\frac { 1 } { 7 } a ^ { 2 } - \frac { 1 } { 7 } a b + \frac { 3 } { 49 } ( a - b ) ^ { 2 } - \frac { 6 } { 49 } a ^ { 2 } + \frac { 6 } { 49 } a b$$

Evaluate

$$\frac{\left(a-b\right)\left(4a-3b\right)}{49}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$\frac{1}{7}a^{2}-\frac{1}{7}ab+\frac{3}{49}\left(a^{2}-2ab+b^{2}\right)-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Use the distributive property to multiply \frac{3}{49} by a^{2}-2ab+b^{2}.

$$\frac{1}{7}a^{2}-\frac{1}{7}ab+\frac{3}{49}a^{2}-\frac{6}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine \frac{1}{7}a^{2} and \frac{3}{49}a^{2} to get \frac{10}{49}a^{2}.

$$\frac{10}{49}a^{2}-\frac{1}{7}ab-\frac{6}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine -\frac{1}{7}ab and -\frac{6}{49}ab to get -\frac{13}{49}ab.

$$\frac{10}{49}a^{2}-\frac{13}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine \frac{10}{49}a^{2} and -\frac{6}{49}a^{2} to get \frac{4}{49}a^{2}.

$$\frac{4}{49}a^{2}-\frac{13}{49}ab+\frac{3}{49}b^{2}+\frac{6}{49}ab$$

Combine -\frac{13}{49}ab and \frac{6}{49}ab to get -\frac{1}{7}ab.

$$\frac{4}{49}a^{2}-\frac{1}{7}ab+\frac{3}{49}b^{2}$$

Expand

$$-\frac{ab}{7}+\frac{3b^{2}}{49}+\frac{4a^{2}}{49}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$\frac{1}{7}a^{2}-\frac{1}{7}ab+\frac{3}{49}\left(a^{2}-2ab+b^{2}\right)-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Use the distributive property to multiply \frac{3}{49} by a^{2}-2ab+b^{2}.

$$\frac{1}{7}a^{2}-\frac{1}{7}ab+\frac{3}{49}a^{2}-\frac{6}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine \frac{1}{7}a^{2} and \frac{3}{49}a^{2} to get \frac{10}{49}a^{2}.

$$\frac{10}{49}a^{2}-\frac{1}{7}ab-\frac{6}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine -\frac{1}{7}ab and -\frac{6}{49}ab to get -\frac{13}{49}ab.

$$\frac{10}{49}a^{2}-\frac{13}{49}ab+\frac{3}{49}b^{2}-\frac{6}{49}a^{2}+\frac{6}{49}ab$$

Combine \frac{10}{49}a^{2} and -\frac{6}{49}a^{2} to get \frac{4}{49}a^{2}.

$$\frac{4}{49}a^{2}-\frac{13}{49}ab+\frac{3}{49}b^{2}+\frac{6}{49}ab$$

Combine -\frac{13}{49}ab and \frac{6}{49}ab to get -\frac{1}{7}ab.

$$\frac{4}{49}a^{2}-\frac{1}{7}ab+\frac{3}{49}b^{2}$$