Solve (2a/3+b/2)(b/2-2a/3)

$$( \frac { 2 a } { 3 } + \frac { b } { 2 } ) ( \frac { b } { 2 } - \frac { 2 a } { 3 } )$$

Evaluate

$$\frac{b^{2}}{4}-\frac{4a^{2}}{9}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2a}{3} times \frac{2}{2}. Multiply \frac{b}{2} times \frac{3}{3}.

$$\left(\frac{2\times 2a}{6}+\frac{3b}{6}\right)\left(\frac{b}{2}-\frac{2a}{3}\right)$$

Since \frac{2\times 2a}{6} and \frac{3b}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2a+3b}{6}\left(\frac{b}{2}-\frac{2a}{3}\right)$$

Do the multiplications in 2\times 2a+3b.

$$\frac{4a+3b}{6}\left(\frac{b}{2}-\frac{2a}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{b}{2} times \frac{3}{3}. Multiply \frac{2a}{3} times \frac{2}{2}.

$$\frac{4a+3b}{6}\left(\frac{3b}{6}-\frac{2\times 2a}{6}\right)$$

Since \frac{3b}{6} and \frac{2\times 2a}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4a+3b}{6}\times \frac{3b-2\times 2a}{6}$$

Do the multiplications in 3b-2\times 2a.

$$\frac{4a+3b}{6}\times \frac{3b-4a}{6}$$

Multiply \frac{4a+3b}{6} times \frac{3b-4a}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4a+3b\right)\left(3b-4a\right)}{6\times 6}$$

Multiply 6 and 6 to get 36.

$$\frac{\left(4a+3b\right)\left(3b-4a\right)}{36}$$

Consider \left(4a+3b\right)\left(3b-4a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3b\right)^{2}-\left(4a\right)^{2}}{36}$$

Expand \left(3b\right)^{2}.

$$\frac{3^{2}b^{2}-\left(4a\right)^{2}}{36}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9b^{2}-\left(4a\right)^{2}}{36}$$

Expand \left(4a\right)^{2}.

$$\frac{9b^{2}-4^{2}a^{2}}{36}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{9b^{2}-16a^{2}}{36}$$

Expand

$$\frac{b^{2}}{4}-\frac{4a^{2}}{9}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 2 is 6. Multiply \frac{2a}{3} times \frac{2}{2}. Multiply \frac{b}{2} times \frac{3}{3}.

$$\left(\frac{2\times 2a}{6}+\frac{3b}{6}\right)\left(\frac{b}{2}-\frac{2a}{3}\right)$$

Since \frac{2\times 2a}{6} and \frac{3b}{6} have the same denominator, add them by adding their numerators.

$$\frac{2\times 2a+3b}{6}\left(\frac{b}{2}-\frac{2a}{3}\right)$$

Do the multiplications in 2\times 2a+3b.

$$\frac{4a+3b}{6}\left(\frac{b}{2}-\frac{2a}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{b}{2} times \frac{3}{3}. Multiply \frac{2a}{3} times \frac{2}{2}.

$$\frac{4a+3b}{6}\left(\frac{3b}{6}-\frac{2\times 2a}{6}\right)$$

Since \frac{3b}{6} and \frac{2\times 2a}{6} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4a+3b}{6}\times \frac{3b-2\times 2a}{6}$$

Do the multiplications in 3b-2\times 2a.

$$\frac{4a+3b}{6}\times \frac{3b-4a}{6}$$

Multiply \frac{4a+3b}{6} times \frac{3b-4a}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(4a+3b\right)\left(3b-4a\right)}{6\times 6}$$

Multiply 6 and 6 to get 36.

$$\frac{\left(4a+3b\right)\left(3b-4a\right)}{36}$$

Consider \left(4a+3b\right)\left(3b-4a\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3b\right)^{2}-\left(4a\right)^{2}}{36}$$

Expand \left(3b\right)^{2}.

$$\frac{3^{2}b^{2}-\left(4a\right)^{2}}{36}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9b^{2}-\left(4a\right)^{2}}{36}$$

Expand \left(4a\right)^{2}.

$$\frac{9b^{2}-4^{2}a^{2}}{36}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{9b^{2}-16a^{2}}{36}$$