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Quadratic Equation: Discriminants and Complex Numbers

When using the quadratic formula to solve a quadratic equation ax2 + bx + c = 0, the discriminant is b2 - 4ac. This discriminant can be positive, zero, or negative. (When the discriminate is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt(-1) = i. )

Explain what the value of the discriminant means to the graph of y = ax2 + bx + c. Hint: Chose values of a, b and c to create a particular discriminant. Then, graph the corresponding equation.

What are examples of quadratic equations or applications with imaginary numbers?

Solution Preview

So we consider the quadratic function y=ax^2+bx+c.

When a>0, the parabola is concave up and has a minimum turning point when x=-b/2a.
When a>0, the parabola is concave down and has a maximum turning point at x=-b/2a.

The intersection points (if any) with the x-axis occur when y=0, so when ax^2+bx+c=0.

When a>0 so the parabola curves upwards.
In this case, if the disciminant is positive, this means that the intersection of the parabola with the x-axis gives two solutions, so it crosses the x-axis twice.
If the disciminant is zero, this means that the intersection of the parabola with the x-axis gives precisely one solution, so it just touches the x-axis.
If the disciminant is negative, the euqation ax^2+bx+c=0 has no real solutions (only imaginary ones) so this means that the parabola does not cross the x=axis at all. (it is what they call positive definite)

So to summarise the situation. Let's denote the discriminant by D = b^2-4ac

(1)(i) a>0, D>0 - the parabola curves up, and crosses the x-axis in two places
(1)(ii) a>0, D=0 - the parabola curves up, and touches the x-axis in exactly one ...

Solution Summary

Quadratic equations, discriminants and imaginary (complex) numbers are investigated in this solution, which is detailed and well presented. Examples and references are included.

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