Optimization

The optimization of a quadratic equation is the process to find the maximum or minimum of said quadratic.

Process

It involves converting a quadratic to the standard form $a(x-h)^2+k$ by completing the square. Then by the Trivial Inequality, the maximum or minimum (it depends on which way the graph of the quadratic is facing) is $k$ .

"Formula"

To optimize a quadratic, one might use the method described above, or one could use this other, smoother, method:

If $a>0$ , then the quadratic $ax^2+bx+c=0$ reaches its minimum when $x=-\frac{b}{2a}$ , while when $a<0$ , the quadratic reaches its maximum when $x=-\frac{b}{2a}$ .

Alternative

The alternative requires calculus. The optimum point has a tangent line with a slope of 0. Thus, calculating the derivative and setting it equal to 0 will also give an answer. Assuming that the original quadratic is $ax^2+bx+c$ , the derivative is $2ax+b$ . Setting $2ax+b = 0$ , we find that $x = -\frac{b}{2a}$ .

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Optimization

Process

"Formula"

Alternative