Solve (2/3 * p + 1/2 * q)(2/5 * p - 4/5 * q) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$(\frac{2}{3}p+\frac{1}{2}q)(\frac{2}{5}p-\frac{4}{5}q)$$

Evaluate

$\frac{\left(p-2q\right)\left(4p+3q\right)}{15}$

Solution Steps

Apply the distributive property by multiplying each term of $\frac{2}{3}p+\frac{1}{2}q$ by each term of $\frac{2}{5}p-\frac{4}{5}q$.

$$\frac{2}{3}p\times \frac{2}{5}p+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q\left(-\frac{4}{5}\right)q$$

Multiply $p$ and $p$ to get $p^{2}$.

$$\frac{2}{3}p^{2}\times \frac{2}{5}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q\left(-\frac{4}{5}\right)q$$

Multiply $q$ and $q$ to get $q^{2}$.

$$\frac{2}{3}p^{2}\times \frac{2}{5}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{2}{3}$ times $\frac{2}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 2}{3\times 5}p^{2}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Do the multiplications in the fraction $\frac{2\times 2}{3\times 5}$.

$$\frac{4}{15}p^{2}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{2}{3}$ times $-\frac{4}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}+\frac{2\left(-4\right)}{3\times 5}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Do the multiplications in the fraction $\frac{2\left(-4\right)}{3\times 5}$.

$$\frac{4}{15}p^{2}+\frac{-8}{15}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Fraction $\frac{-8}{15}$ can be rewritten as $-\frac{8}{15}$ by extracting the negative sign.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{1}{2}$ times $\frac{2}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1\times 2}{2\times 5}qp+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Cancel out $2$ in both numerator and denominator.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1}{5}qp+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Combine $-\frac{8}{15}pq$ and $\frac{1}{5}qp$ to get $-\frac{1}{3}pq$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{1}{2}$ times $-\frac{4}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{1\left(-4\right)}{2\times 5}q^{2}$$

Do the multiplications in the fraction $\frac{1\left(-4\right)}{2\times 5}$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{-4}{10}q^{2}$$

Reduce the fraction $\frac{-4}{10}$ to lowest terms by extracting and canceling out $2$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq-\frac{2}{5}q^{2}$$

Expand

$-\frac{pq}{3}+\frac{4p^{2}}{15}-\frac{2q^{2}}{5}$

Solution Steps

Apply the distributive property by multiplying each term of $\frac{2}{3}p+\frac{1}{2}q$ by each term of $\frac{2}{5}p-\frac{4}{5}q$.

$$\frac{2}{3}p\times \frac{2}{5}p+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q\left(-\frac{4}{5}\right)q$$

Multiply $p$ and $p$ to get $p^{2}$.

$$\frac{2}{3}p^{2}\times \frac{2}{5}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q\left(-\frac{4}{5}\right)q$$

Multiply $q$ and $q$ to get $q^{2}$.

$$\frac{2}{3}p^{2}\times \frac{2}{5}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{2}{3}$ times $\frac{2}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\times 2}{3\times 5}p^{2}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Do the multiplications in the fraction $\frac{2\times 2}{3\times 5}$.

$$\frac{4}{15}p^{2}+\frac{2}{3}p\left(-\frac{4}{5}\right)q+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{2}{3}$ times $-\frac{4}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}+\frac{2\left(-4\right)}{3\times 5}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Do the multiplications in the fraction $\frac{2\left(-4\right)}{3\times 5}$.

$$\frac{4}{15}p^{2}+\frac{-8}{15}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Fraction $\frac{-8}{15}$ can be rewritten as $-\frac{8}{15}$ by extracting the negative sign.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1}{2}q\times \frac{2}{5}p+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{1}{2}$ times $\frac{2}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1\times 2}{2\times 5}qp+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Cancel out $2$ in both numerator and denominator.

$$\frac{4}{15}p^{2}-\frac{8}{15}pq+\frac{1}{5}qp+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Combine $-\frac{8}{15}pq$ and $\frac{1}{5}qp$ to get $-\frac{1}{3}pq$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{1}{2}q^{2}\left(-\frac{4}{5}\right)$$

Multiply $\frac{1}{2}$ times $-\frac{4}{5}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{1\left(-4\right)}{2\times 5}q^{2}$$

Do the multiplications in the fraction $\frac{1\left(-4\right)}{2\times 5}$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq+\frac{-4}{10}q^{2}$$

Reduce the fraction $\frac{-4}{10}$ to lowest terms by extracting and canceling out $2$.

$$\frac{4}{15}p^{2}-\frac{1}{3}pq-\frac{2}{5}q^{2}$$