Solve (2x-3/5y-1/3z)(3/5y-1/3z+2x)

$$( 2 x - \frac { 3 } { 5 } y - \frac { 1 } { 3 } z ) ( \frac { 3 } { 5 } y - \frac { 1 } { 3 } z + 2 x )$$

Evaluate

$$\frac{\left(z-6x\right)^{2}}{9}-\frac{9y^{2}}{25}$$

Solution Steps

Apply the distributive property by multiplying each term of 2x-\frac{3}{5}y-\frac{1}{3}z by each term of \frac{3}{5}y-\frac{1}{3}z+2x.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y\times \frac{3}{5}y-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z\left(-\frac{1}{3}\right)z-\frac{1}{3}z\times 2x$$

Multiply y and y to get y^{2}.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z\left(-\frac{1}{3}\right)z-\frac{1}{3}z\times 2x$$

Multiply z and z to get z^{2}.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express 2\times \frac{3}{5} as a single fraction.

$$\frac{2\times 3}{5}xy+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply 2 and 3 to get 6.

$$\frac{6}{5}xy+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express 2\left(-\frac{1}{3}\right) as a single fraction.

$$\frac{6}{5}xy+\frac{2\left(-1\right)}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply 2 and -1 to get -2.

$$\frac{6}{5}xy+\frac{-2}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

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

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}+\frac{-3\times 3}{5\times 5}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

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

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}+\frac{-9}{25}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-9}{25} can be rewritten as -\frac{9}{25} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{3}{5} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{-3\left(-1\right)}{5\times 3}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Do the multiplications in the fraction \frac{-3\left(-1\right)}{5\times 3}.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{3}{15}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Reduce the fraction \frac{3}{15} to lowest terms by extracting and canceling out 3.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express -\frac{3}{5}\times 2 as a single fraction.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-3\times 2}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -3 and 2 to get -6.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-6}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-6}{5} can be rewritten as -\frac{6}{5} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{6}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Combine \frac{6}{5}xy and -\frac{6}{5}yx to get 0.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{1}{3} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-3}{3\times 5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Cancel out 3 in both numerator and denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-1}{5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-1}{5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{1}{5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Combine \frac{1}{5}yz and -\frac{1}{5}zy to get 0.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{1}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{-\left(-1\right)}{3\times 3}z^{2}-\frac{1}{3}z\times 2x$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{3\times 3}.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}-\frac{1}{3}z\times 2x$$

Express -\frac{1}{3}\times 2 as a single fraction.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}+\frac{-2}{3}zx$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}-\frac{2}{3}zx$$

Combine -\frac{2}{3}xz and -\frac{2}{3}zx to get -\frac{4}{3}xz.

$$-\frac{4}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}$$

Expand

$$-\frac{4xz}{3}+\frac{z^{2}}{9}-\frac{9y^{2}}{25}+4x^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of 2x-\frac{3}{5}y-\frac{1}{3}z by each term of \frac{3}{5}y-\frac{1}{3}z+2x.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y\times \frac{3}{5}y-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z\left(-\frac{1}{3}\right)z-\frac{1}{3}z\times 2x$$

Multiply y and y to get y^{2}.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z\left(-\frac{1}{3}\right)z-\frac{1}{3}z\times 2x$$

Multiply z and z to get z^{2}.

$$2x\times \frac{3}{5}y+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express 2\times \frac{3}{5} as a single fraction.

$$\frac{2\times 3}{5}xy+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply 2 and 3 to get 6.

$$\frac{6}{5}xy+2x\left(-\frac{1}{3}\right)z+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express 2\left(-\frac{1}{3}\right) as a single fraction.

$$\frac{6}{5}xy+\frac{2\left(-1\right)}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply 2 and -1 to get -2.

$$\frac{6}{5}xy+\frac{-2}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{3}{5}y^{2}\times \frac{3}{5}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

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

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}+\frac{-3\times 3}{5\times 5}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

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

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}+\frac{-9}{25}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-9}{25} can be rewritten as -\frac{9}{25} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}-\frac{3}{5}y\left(-\frac{1}{3}\right)z-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{3}{5} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{-3\left(-1\right)}{5\times 3}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Do the multiplications in the fraction \frac{-3\left(-1\right)}{5\times 3}.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{3}{15}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Reduce the fraction \frac{3}{15} to lowest terms by extracting and canceling out 3.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{3}{5}y\times 2x-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Express -\frac{3}{5}\times 2 as a single fraction.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-3\times 2}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -3 and 2 to get -6.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-6}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-6}{5} can be rewritten as -\frac{6}{5} by extracting the negative sign.

$$\frac{6}{5}xy-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{6}{5}yx-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Combine \frac{6}{5}xy and -\frac{6}{5}yx to get 0.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{1}{3}z\times \frac{3}{5}y-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{1}{3} times \frac{3}{5} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-3}{3\times 5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Cancel out 3 in both numerator and denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz+\frac{-1}{5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Fraction \frac{-1}{5} can be rewritten as -\frac{1}{5} by extracting the negative sign.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{5}yz-\frac{1}{5}zy-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Combine \frac{1}{5}yz and -\frac{1}{5}zy to get 0.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}-\frac{1}{3}z^{2}\left(-\frac{1}{3}\right)-\frac{1}{3}z\times 2x$$

Multiply -\frac{1}{3} times -\frac{1}{3} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{-\left(-1\right)}{3\times 3}z^{2}-\frac{1}{3}z\times 2x$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{3\times 3}.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}-\frac{1}{3}z\times 2x$$

Express -\frac{1}{3}\times 2 as a single fraction.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}+\frac{-2}{3}zx$$

Fraction \frac{-2}{3} can be rewritten as -\frac{2}{3} by extracting the negative sign.

$$-\frac{2}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}-\frac{2}{3}zx$$

Combine -\frac{2}{3}xz and -\frac{2}{3}zx to get -\frac{4}{3}xz.

$$-\frac{4}{3}xz+4x^{2}-\frac{9}{25}y^{2}+\frac{1}{9}z^{2}$$