Solve quad4(a+2b)(a-2b)-1/2b(a-8b)

$$\quad 4 ( a + 2 b ) ( a - 2 b ) - \frac { 1 } { 2 } b ( a - 8 b )$$

Evaluate

$$-\frac{ab}{2}+4a^{2}-12b^{2}$$

Solution Steps

Use the distributive property to multiply 4 by a+2b.

$$\left(4a+8b\right)\left(a-2b\right)-\frac{1}{2}b\left(a-8b\right)$$

Apply the distributive property by multiplying each term of 4a+8b by each term of a-2b.

$$4a^{2}-8ab+8ba-16b^{2}-\frac{1}{2}b\left(a-8b\right)$$

Combine -8ab and 8ba to get 0.

$$4a^{2}-16b^{2}-\frac{1}{2}b\left(a-8b\right)$$

Use the distributive property to multiply -\frac{1}{2}b by a-8b.

$$4a^{2}-16b^{2}-\frac{1}{2}ba-\frac{1}{2}b\left(-8\right)b$$

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

$$4a^{2}-16b^{2}-\frac{1}{2}ba-\frac{1}{2}b^{2}\left(-8\right)$$

Express -\frac{1}{2}\left(-8\right) as a single fraction.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+\frac{-\left(-8\right)}{2}b^{2}$$

Multiply -1 and -8 to get 8.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+\frac{8}{2}b^{2}$$

Divide 8 by 2 to get 4.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+4b^{2}$$

Combine -16b^{2} and 4b^{2} to get -12b^{2}.

$$4a^{2}-12b^{2}-\frac{1}{2}ba$$

Expand

$$-\frac{ab}{2}+4a^{2}-12b^{2}$$

Solution Steps

Use the distributive property to multiply 4 by a+2b.

$$\left(4a+8b\right)\left(a-2b\right)-\frac{1}{2}b\left(a-8b\right)$$

Apply the distributive property by multiplying each term of 4a+8b by each term of a-2b.

$$4a^{2}-8ab+8ba-16b^{2}-\frac{1}{2}b\left(a-8b\right)$$

Combine -8ab and 8ba to get 0.

$$4a^{2}-16b^{2}-\frac{1}{2}b\left(a-8b\right)$$

Use the distributive property to multiply -\frac{1}{2}b by a-8b.

$$4a^{2}-16b^{2}-\frac{1}{2}ba-\frac{1}{2}b\left(-8\right)b$$

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

$$4a^{2}-16b^{2}-\frac{1}{2}ba-\frac{1}{2}b^{2}\left(-8\right)$$

Express -\frac{1}{2}\left(-8\right) as a single fraction.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+\frac{-\left(-8\right)}{2}b^{2}$$

Multiply -1 and -8 to get 8.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+\frac{8}{2}b^{2}$$

Divide 8 by 2 to get 4.

$$4a^{2}-16b^{2}-\frac{1}{2}ba+4b^{2}$$

Combine -16b^{2} and 4b^{2} to get -12b^{2}.

$$4a^{2}-12b^{2}-\frac{1}{2}ba$$