+1839 If $s$ is a real number, then what is the smallest possible value of $2s^2 - 8s + 19 - 7s^2 - 8s + 20$?
+9673 I assume you meant largest.
Simplifying the expression and completing the square, we have
\(2s^2 - 8s + 19 - 7s^2 - 8s + 20\\ = -5s^2 - 16s + 39\\ = -5(s^2 + \frac{16}5s) + 39\\ = -5(s + \frac85)^2 + 5(\frac 85)^2 + 39\\ =-5(s + \frac85)^2 + \frac{259}5\)
Note that \(-5(s + \frac85)^2 \leq 0\) for any real s. Then the largest possible value is \(\dfrac{259}5\).
There is no smallest value. The expression can be arbitrarily small.
+9673 I assume you meant largest.
Simplifying the expression and completing the square, we have
\(2s^2 - 8s + 19 - 7s^2 - 8s + 20\\ = -5s^2 - 16s + 39\\ = -5(s^2 + \frac{16}5s) + 39\\ = -5(s + \frac85)^2 + 5(\frac 85)^2 + 39\\ =-5(s + \frac85)^2 + \frac{259}5\)
Note that \(-5(s + \frac85)^2 \leq 0\) for any real s. Then the largest possible value is \(\dfrac{259}5\).
There is no smallest value. The expression can be arbitrarily small.
