Solve (3z-1/2+b)(3z-b-1/2)

$$( 3 z - \frac { 1 } { 2 } + b ) ( 3 z - b - \frac { 1 } { 2 } )$$

Evaluate

$$\frac{\left(6z-1\right)^{2}}{4}-b^{2}$$

Solution Steps

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

$$9z^{2}-3zb+3z\left(-\frac{1}{2}\right)-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb+\frac{3\left(-1\right)}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Multiply 3 and -1 to get -3.

$$9z^{2}-3zb+\frac{-3}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z+\frac{-3}{2}z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z-\frac{3}{2}z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine -\frac{3}{2}z and -\frac{3}{2}z to get -3z.

$$9z^{2}-3zb-3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$9z^{2}-3zb-3z+\frac{1}{2}b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-3z+\frac{1}{2}b+\frac{-\left(-1\right)}{2\times 2}+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-3z+\frac{1}{2}b+\frac{1}{4}+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine -3zb and 3bz to get 0.

$$9z^{2}-3z+\frac{1}{2}b+\frac{1}{4}-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}b and b\left(-\frac{1}{2}\right) to get 0.

$$9z^{2}-3z+\frac{1}{4}-b^{2}$$

Expand

$$9z^{2}-3z-b^{2}+\frac{1}{4}$$

Solution Steps

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

$$9z^{2}-3zb+3z\left(-\frac{1}{2}\right)-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb+\frac{3\left(-1\right)}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Multiply 3 and -1 to get -3.

$$9z^{2}-3zb+\frac{-3}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z-\frac{1}{2}\times 3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z+\frac{-3}{2}z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-\frac{3}{2}z-\frac{3}{2}z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine -\frac{3}{2}z and -\frac{3}{2}z to get -3z.

$$9z^{2}-3zb-3z-\frac{1}{2}\left(-1\right)b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Multiply -\frac{1}{2} and -1 to get \frac{1}{2}.

$$9z^{2}-3zb-3z+\frac{1}{2}b-\frac{1}{2}\left(-\frac{1}{2}\right)+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-3z+\frac{1}{2}b+\frac{-\left(-1\right)}{2\times 2}+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

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

$$9z^{2}-3zb-3z+\frac{1}{2}b+\frac{1}{4}+3bz-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine -3zb and 3bz to get 0.

$$9z^{2}-3z+\frac{1}{2}b+\frac{1}{4}-b^{2}+b\left(-\frac{1}{2}\right)$$

Combine \frac{1}{2}b and b\left(-\frac{1}{2}\right) to get 0.

$$9z^{2}-3z+\frac{1}{4}-b^{2}$$