Explain the meaning of each term to the right of the equal, in terms of a graph.

Question

X=-b/2a+/-√b^2-4ac/2a

Answer 1

1) x = (-b ± √(b²-4ac))2a

are the two solutions to the quadratic equation:

2) ax²+bx+c=0

when a, b, and c are known, two solutions are found by using eqn 1) one using a + sign and one using a - sign where you see the ± sign.

There is another way of looking at this. If we consider the graph of:

3) y = ax²+bx+c

This is the general equation for a parabola, which either crosses the x-axis twice or not at all.

Picture the graph of a parabola, it can concave up (smiling), or concave down (frowning), and it can be drawn completely above or below the x-axis, or it can be drawn crossing the x-axis in two places. These are the only choices. Go ahead and draw some freehand on graph paper right now to see what I mean.

Now,

If:

(b²-4ac) > 0

then the square root in eqn 1) is real and the two solutions from eqn 1) yield the x-intercepts of the parabola, that is where the parabola crosses the x-axis, which will be in two places.

If:

(b²-4ac) < 0

then the square root in eqn 1) is complex and the equation has no real roots, thus, in this case, the parabola does not cross the x-axis, its graph is either entirely above or below the x-axis.