Solve left(a+2bright)left(a-2bright)-1/2bleft(a-8bright)

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

Evaluate

$$-\frac{ab}{2}+a^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Multiply -1 and -8 to get 8.

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

Divide 8 by 2 to get 4.

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

Combine -4b^{2} and 4b^{2} to get 0.

$$a^{2}-\frac{1}{2}ba$$

Expand

$$-\frac{ab}{2}+a^{2}$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

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

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

Multiply -1 and -8 to get 8.

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

Divide 8 by 2 to get 4.

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

Combine -4b^{2} and 4b^{2} to get 0.

$$a^{2}-\frac{1}{2}ba$$