Solve left(a+cright)^2-left(a-cright)^2

$${ \left(a+c \right) }^{ 2 } - { \left(a-c \right) }^{ 2 }$$

Evaluate

$$4ac$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+c\right)^{2}.

$$a^{2}+2ac+c^{2}-\left(a-c\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-c\right)^{2}.

$$a^{2}+2ac+c^{2}-\left(a^{2}-2ac+c^{2}\right)$$

To find the opposite of a^{2}-2ac+c^{2}, find the opposite of each term.

$$a^{2}+2ac+c^{2}-a^{2}+2ac-c^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$2ac+c^{2}+2ac-c^{2}$$

Combine 2ac and 2ac to get 4ac.

$$4ac+c^{2}-c^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$4ac$$

Expand

$$4ac$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+c\right)^{2}.

$$a^{2}+2ac+c^{2}-\left(a-c\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-c\right)^{2}.

$$a^{2}+2ac+c^{2}-\left(a^{2}-2ac+c^{2}\right)$$

To find the opposite of a^{2}-2ac+c^{2}, find the opposite of each term.

$$a^{2}+2ac+c^{2}-a^{2}+2ac-c^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$2ac+c^{2}+2ac-c^{2}$$

Combine 2ac and 2ac to get 4ac.

$$4ac+c^{2}-c^{2}$$

Combine c^{2} and -c^{2} to get 0.

$$4ac$$