+1839 What real value of $t$ produces the smallest value of the quadratic $t^2 -9t - 36 + 13t - 60$?
+9673 Simplifying and completing the square, we have
\(t^2 -9t - 36 + 13t - 60\\ = t^2 + 4t - 96\\ = t^2 + 4t + 4 - 100\\ = (t + 2)^2 - 100\)
Note that \((t + 2)^2 \geq 0\) for any real t, and equality holds when \(t + 2 = 0\), i.e., \(t = -2\).
Hence, the smallest value is produced when t = -2.
+9673 Simplifying and completing the square, we have
\(t^2 -9t - 36 + 13t - 60\\ = t^2 + 4t - 96\\ = t^2 + 4t + 4 - 100\\ = (t + 2)^2 - 100\)
Note that \((t + 2)^2 \geq 0\) for any real t, and equality holds when \(t + 2 = 0\), i.e., \(t = -2\).
Hence, the smallest value is produced when t = -2.
