Let $r$ and $s$ be the roots of $2x^2 + 5x - 13 = x^2 - 4x + 7.$ Find $r^2 + s^2.$

dpIostthegame Jun 4, 2024

#1**+1 **

First, let's write a quadratic equation to find the two roots.

We have \(x^{2}+9x-20=0\)

Notice that \(r^2 + s^2 = (r+s)^2-2rs\).

Now, let's take a break from this. If we had roots r and s, we would have

\((x-r)(x-s)\\ x^2-sx-rx+rs\\ x^2-(r+s)x+rs\)

Using the equation from above, we find that, r+s is equal to -9 and rs is equal to -20.

Plugging that in to \( (r+s)^2-2rs\), we get\((-9)^2-2(-20) = 81+40 = 121\)

So 121 is our answer!

Thanks! :)\(\)

NotThatSmart Jun 4, 2024

#1**+1 **

Best Answer

First, let's write a quadratic equation to find the two roots.

We have \(x^{2}+9x-20=0\)

Notice that \(r^2 + s^2 = (r+s)^2-2rs\).

Now, let's take a break from this. If we had roots r and s, we would have

\((x-r)(x-s)\\ x^2-sx-rx+rs\\ x^2-(r+s)x+rs\)

Using the equation from above, we find that, r+s is equal to -9 and rs is equal to -20.

Plugging that in to \( (r+s)^2-2rs\), we get\((-9)^2-2(-20) = 81+40 = 121\)

So 121 is our answer!

Thanks! :)\(\)

NotThatSmart Jun 4, 2024