Note that a quadratic with no real solution could have a graph that is completely below the x axis (in that case, one of the conditions is that a < 0).Īs you can see in the graph pictured above, the parabola does not touch the x axis. This means the quadratic equation has no real solution. The vertex of a parabola is sort of like the “mountain top” (for negative values of a) or “valley bottom” (for positive values of a).Īs you can see, the quadratic function f(x) = x 2 – 2x + 2 is a parabola that never touches the x axis, since the vertex (valley bottom) lies above x axis at x = 1. Remember that one of the key features of a parabola is its vertex. For any quadratic equation, the graph will be a parabola. Look At The GraphĪnother way to tell if a quadratic has no real solution is to look at its graph. You can also verify these two solutions by using FOIL on (x – (-1 + 2i))(x – (-1 – 2i)) to get back the original quadratic, x 2 + 2x + 5. (You can verify these solutions by using the quadratic formula with a = 1, b = 2, and c = 5). However, the equation does have two complex solutions, -1 + 2i and -1 – 2i. This means that the quadratic has no real solution. So, b 2 < 4ac (since 4 < 20), and thus the discriminant is negative. Here is one example of a quadratic equation with no real solution:

$quadratic equation examples$

The discriminant is the expression under the radical sign in the quadratic formula.

Quadratic equation examples