y = x^2 + 12x + 5
In the form ax^2 + bx + c.....the x value that minimizes y is given by -b / (2a)
So.....the x value that minimizes y is given by -12 / (2 * 1) = -12 / 2 = -6
Put this into the function and we get that y = (-6)^2 + 12(-6) + 5 = 36 - 72 + 5 = -36 + 5 = -31
What is the minimum possible value for y:
y = x^2 + 12x + 5
\(\begin{array}{|lrcll|} \hline & y &=& x^2 + 12x + 5 \\ & &=& (x+6)^2 -36+5 \\ & &=& (x+6)^2 -31 \qquad \text{min. if } x = -6 \\ & y &=& 0 -31 \\ & y &=& -31 \\ \hline \end{array}\)
The smallest value of the expression \(y= x^2 + 12x + 5\) is -31