where might these imaginary numbers be used?

Question

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.

Answer 1

The discriminat of the equation ax^2 + bx + c determines the number and types of root you will get. There are 3 cases:
1)discriminant > 0 - you will have 2 distinct real roots
2)disctrimnant = 0 - you have 3 identical roots
3)discriminant < 0 - you have 2 distinct imaginery roots

Answer 2

The graph y = ax^2+bx+c can be expressed as:

y = a((x+b/2a)^2 - ((b^2-4ac)/4a)

Setting y = 0 leads straight to the quadratic formula. The discriminant is b^2-4ac.

Geometrically, the fact that x has a b/2a added to it means the graph is displaced to the left by b/2a, so that the minimum of it is at -b/2a. Substituting this value erases the first term, so the y value at this minimum is (b^2-4ac)/4a. Finally the a tells how wide or narrow the graph (a parabola) is.

So the discriminant divided by 4a gives the amount by which the parabola is displaced downwards. If it is negative, the parabola is displaced upwards.

To graph the equation, find the bottom point of the parabola. We have just seen that it is (-b/2a, -D/4a), where D = b^2-4ac is the discriminant. The coefficent a of the x^2 term then tells how wide or narrow the parabola, but simply draw the parabola so it goes through (0,c), as plugging in 0 for x yields c.

Answer 3

All the answers are incorrect so I'll just fix the first one:

The discriminant of the equation ax^2 + bx + c determines the number and types of roots. There are 3 cases:
1)discriminant > 0 implies 2 distinct real roots
2)disctrimnant = 0 implies 2 identical roots
3)discriminant < 0 implies 2 distinct complex roots.

I noticed some answerers talking about a 'determinant' - I think they meant discriminant.

Answer 4

A)+

x^2 + 3x + 1 = y
Determinant = 3^2 - 4(1)(1) = 5, meaning that there are 2 real solutions. On the graph, that means that the minimum of the parabola is below zero and it intersects the x axis at two places.

B)-

x^2 + x + 1 = y
Determinant = 1^2 - 4(1)(1) = -3, meaning that there are 2 complex (not real) solutions. On the graph, that means that all of the parabola is above the x axis.

C)0

x^2 + 2x + 1 = y
Determinant = 2^2 - 4(1)(1) = 0, which means that there is only 1 real solution. On the graph, that means that the minimum of the parabola is right on the x axis.

Answer 5

In the first answer, "case 2" is wrong. You cannot have 3 roots to a quadratic equation, ever. As for the discriminant, it has something to do with the offset from the x value of the vertex where both of the zeros meet the x axis. That's why the zeros are (-b (+/-) sqrt(b^2-4ac))/(2a). I forgot how imaginary numbers play into the discriminant's meaning, but it has something to do with "imaginary" number space. Good luck!

Answer 6

The first three answers are all good answers (except for a typo giving 3 identical answers where the discriminant is 0).

As for what it could be used for:

If you have a differential equation describing harmonic motion, a positive discriminant means the motion is overdamped. The motion quickly damps itself out (just like a good shock absorber on your car).

A discriminant of zero is critically damped.

A negative discriminant means underdamped motion (like a very bad shock absorber on your car).