Suppose a quadratic equation has been given where the a value (ax^2 + bx + c) is a positive and it has been said that the graph of the equation lies above the x-axis- what is the discriminant?
For example- 2x^2 + kx - 5 = y; the graph lies above the x-axis- find the possible values of k. I know how to solve it, just not sure what discriminant to take.
In the same sense, if the a value is a negative, and the graph is said to be lying above the x axis, is the discriminant > 0 ?
On
If the graph of a quadratic function of $x$ lies above or below the $x$-axis, in order words, fails to have real roots, then the discriminant must be negative, since in this case the square root would be imaginary.
Thus, in such a case for your function $ax^2+bx+c,$ it is the case that $$b^2-4ac<0.$$
You need the discrimination to be negative so that the quadratic function has no real roots.
$$\Delta < 0 \iff b^2-4ac < 0$$
Therefore, the graph either lies entirely above or below the $x$-axis. This is determined by the value of $a$.
If $a > 0$, then the graph is concave upward and has a minimum. Clearly, no roots and a minimum implies the graph lies entirely above the $x$-axis.
If $a < 0$, then the graph is concave downward and has a maximum. Clearly, no roots and a maximum implies the graph lies entirely below the $x$-axis.
For your example, you have $y = 2x^2+kx-5$, so you use
$$k^2-4(2)(-5) < 0 \iff k^2+40 < 0 \iff k^2 < -40$$
Clearly, the inequality is not true for real values of $k$, since $k^2 \geq 0$, so no value of $k$ allows the entire graph to remain above the $x$-axis.