Let $a$ and $b$ be complex numbers. If $a + b = 1$ and $a^2 + b^2 = 2,$ then what is $a^3 + b^3?$

parmen Aug 14, 2024

#1**+1 **

Let's give this problem a shot.

Now, we know that \((a+b)^2 = a^2+2ab+b^2\)

From the problem, we ALSO know that \((a+b)^2 = 1^2=1\)

Thus, we have the equation \(a^2+2ab+b^2=1\)

Since the problem gives us the value of \(a^2+b^2=2\), we can easily figure out what ab is.

We have

\(2ab+2=1\\ 2ab=-1\\ ab=-1/2\)

The value of ab will come in handy later.

Now, let's focus on what we must find. a^3+b^3. From a handy equation, we know that

\(a^3+b^3=\left(a+b\right)\left(a^2-ab+b^2\right)\)

Wait! We already know all the values needed to solve the problem. Plugging in 1, 2, and -1/2, we get

\(a^3+b^2=(1)(2-(-1/2)) = 2+1/2 = 5/2\)

Thus, our final answer is 5/2.

Thanks! :)

NotThatSmart Aug 15, 2024