Question

1. is ((a + b)^2) equal to (a^2 + b^2)? justify your answer completely

Explanation:

Step1: Expand $(a+b)^2$

$$(a+b)^2=(a+b)(a+b)$$

Step2: Distribute terms via FOIL

$$(a+b)(a+b)=a^2+ab+ab+b^2$$

Step3: Combine like terms

$$a^2+ab+ab+b^2=a^2+2ab+b^2$$

Step4: Compare to $a^2+b^2$

$a^2+2ab+b^2
eq a^2+b^2$ unless $ab=0$

Answer:

No, $(a+b)^2$ is not completely equal to $a^2+b^2$. Expanding $(a+b)^2$ gives $a^2+2ab+b^2$, which includes an additional $2ab$ term. This term is only zero if $a=0$, $b=0$, or both, so the two expressions are only equal in those specific cases, not universally.