+36 Let $r$ and $s$ be the roots of $2x^2 + 5x - 13 = x^2 - 4x + 7.$ Find $r^2 + s^2.$
+1894 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! :)\(\)
+1894 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! :)\(\)
