Question

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!

Answer 1

516

Explanation:

Let's start with the value first to see if we can use
`a^2+b^2=258` to help find it's value:

`(a+b)^2+(a-b)^2`

I'm going to use the formula
`(u+v)^2=u^2+2uv+v^2` to expand both:

`a^2+2ab+b^2+a^2-2ab+b^2`

Combining like terms:

`2a^2+2b^2`

Factoring the 2 out:

`2(a^2+b^2)`

Plug in 258 for the
`a^2+b^2`:

`2(258)`

Perform the multiplication:

`516`

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Another way:

Find values for
`a` and
`b` that satisfy:

`a^2+b^2=258`

The easiest solution you might see is
`a=√(258) \text{ while }b=0`. This works because the square of
`√(258)` is 258.

So now you just plug:

`(a+b)^2+(a-b)^2` with
`a` being
`√(258)` and
`b` being 0 into your calculator or if you are good at simplifying things without you can do that with this problem:

`(√(258)+0)^2+(√(258)-0)^2`

`(√(258))^2+√(258))^2`

`258+258`

`2(258)`

`516`

This would have worked for any pair
`(a,b)` satisfying
`a^2+b^2=258`.

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.