SAT Math Question: If a ^2+b^2=258, what is the value of (a+b)^2+(a-b)^2??Please explain how you arrived at your answer. Thank you!

Accepted Solution

Answer:516Step-by-step explanation:Let's start with the value first to see if we can use [tex]a^2+b^2=258[/tex] to help find it's value:[tex](a+b)^2+(a-b)^2[/tex]I'm going to use the formula [tex](u+v)^2=u^2+2uv+v^2[/tex] to expand both:[tex]a^2+2ab+b^2+a^2-2ab+b^2[/tex]Combining like terms:[tex]2a^2+2b^2[/tex]Factoring the 2 out:[tex]2(a^2+b^2)[/tex]Plug in 258 for the [tex]a^2+b^2[/tex]:[tex]2(258)[/tex]Perform the multiplication:[tex]516[/tex]-----------------------------------------------Another way:Find values for [tex]a[/tex] and [tex]b[/tex] that satisfy:[tex]a^2+b^2=258[/tex]The easiest solution you might see is [tex]a=\sqrt{258} \text{ while }b=0[/tex].  This works because the square of [tex]\sqrt{258}[/tex] is 258.So now you just plug:[tex](a+b)^2+(a-b)^2[/tex] with [tex]a[/tex] being [tex]\sqrt{258}[/tex] and [tex]b[/tex] being 0 into your calculator or if you are good at simplifying things without you can do that with this problem:[tex](\sqrt{258}+0)^2+(\sqrt{258}-0)^2[/tex][tex](\sqrt{258})^2+\sqrt{258})^2[/tex][tex]258+258[/tex][tex]2(258)[/tex][tex]516[/tex]This would have worked for any pair [tex](a,b)[/tex] satisfying [tex]a^2+b^2=258[/tex].I wanted to show this last strategy just in case you haven't been exposed to expanding squared binomials with foil or the formula I mentioned earlier.