Answer:
(a²+b²) = 80 and (a-b)² = 16
Explanation:
Let's first expand out (a-b)^2.
a^2 - 2ab + b^2
(a^2 + b^2) - 2*ab
The goal is to find what a^2 + b^2 is.
Now while we can use system of equations to find the separate values of a and b and then plug it in, there is a more efficient way to solve this problem. See below.
Let's square both sides of the equation (a+b) = 12
(a+b)^2 = 12^2 Now we can expand and simplify this equation
a^2 + b^2 + 2ab = 144
We know the value of ab because it is given so let's substitute that in.
a^2 + b^2 + 2*32 = 144
Now let's solve for a^2 + b^2 by taking 2*32 to the other side and simplifying.
a^2 + b^2 = 144 - 64 = 80
a^2 + b^2 = 80
Now to find the value of (a-b)^2
We expanded this out earlier and found that its a combination of a^2+b^2 and -2ab, both values which we have. So let's substitute in, simplify, and get our answer.
(a^2 + b^2) - 2*ab
80 - 2*32 = 80 - 64 = 16
Hope this helps :)