Solve (2t+1)^2-(4t-5)^2

$$( 2 t + 1 ) ^ { 2 } - ( 4 t - 5 ) ^ { 2 }$$

Evaluate

$$-12t^{2}+44t-24$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+1\right)^{2}.

$$4t^{2}+4t+1-\left(4t-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4t-5\right)^{2}.

$$4t^{2}+4t+1-\left(16t^{2}-40t+25\right)$$

To find the opposite of 16t^{2}-40t+25, find the opposite of each term.

$$4t^{2}+4t+1-16t^{2}+40t-25$$

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

$$-12t^{2}+4t+1+40t-25$$

Combine 4t and 40t to get 44t.

$$-12t^{2}+44t+1-25$$

Subtract 25 from 1 to get -24.

$$-12t^{2}+44t-24$$

Expand

$$-12t^{2}+44t-24$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2t+1\right)^{2}.

$$4t^{2}+4t+1-\left(4t-5\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4t-5\right)^{2}.

$$4t^{2}+4t+1-\left(16t^{2}-40t+25\right)$$

To find the opposite of 16t^{2}-40t+25, find the opposite of each term.

$$4t^{2}+4t+1-16t^{2}+40t-25$$

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

$$-12t^{2}+4t+1+40t-25$$

Combine 4t and 40t to get 44t.

$$-12t^{2}+44t+1-25$$

Subtract 25 from 1 to get -24.

$$-12t^{2}+44t-24$$