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.

The standard quadratic equation is given as: ax^2 + bx + c = 0 , where a, b, and c are real numbers and a > 0. The quadratic formula can be used to solve for all solutions of any quadratic equation. If a graph of the quadratic equation crosses the x axis, it has two real number solutions. We use the discriminant of b^2 - 4ac,

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.
Create three unique equations where the discriminant is positive, zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx +

Compute the value of the discriminant and give the number of real solutions to the quadratic equation.
Please see attached file for full problem description.

If I am 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 discriminant is negative, then we have the square root of a negative number. This is called an imaginary number, sqrt(-1) = i. )
Explain what the val

First Part:
What is the discriminant? Explain what information the discriminant gives us and why this information is important. What does the discriminant tell us about the graph of a quadratic equation?
Second Part:
What is a parabola and where does it come from? What important information is given by the vertex of a p

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.
What I need to do is figure out how to create three unique equations where the discriminant is positive, zero, or negative. For each case, please explain what t

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.
Create three unique equations where the discriminant is positive, zero, or negative. For each case, explain what this value means to the graph of y = ax2 + bx +

Determine if the following equation has a solution or not? Justify your answer.
√2x^2 - 4x - 7√2 = 0
I figured that you have to multiply both sides by 1/2 to cancel the square roots. The equation then becomes x^2 - 2x - 7 = 0.

1. Determine the number of solutions and classify the type of solutions for each of the following equations. Justify your answer.
a) x2 + 3x - 15 = 0
b) x2 + x + 4 = 0
c) x2 - 4x - 7 = 0
d) x2 - 8x + 16 = 0
e) 2x2 - 3x + 7 = 0
f) x2 - 4x - 77 = 0
g) 3x2 - 7x + 6 = 0
h) 4x2 + 16x + 16 = 0
2. Find an equatio