Evaluate the discriminant of each equation. Tell how many solutions each equation has and whether the solutions are real or imaginary.

y=9x^2 - 24x

y=-2x^2 + 3x - 5

y=-4x^2 - 4x - 1

The term b ^ 2 - 4 a c

is called the discriminant and gives important information about the number and nature of the solutions to the quadratic equation to be solved.

Three cases are possible:

If D > 0, the equation has 2 real solutions.

If D = 0, the equation has 1 real solution.

If D < 0, the equation has 2 conjugate imaginary solutions.

In google type:

quadratic equation online

When you see list of results click on:

Free Online Quadratic Equation Solver:Solve by Quadratic Formula

When page be open in rectangle type:

9 x ^ 2 - 24 x = 0

and click option: solve it!

You will see everything about function,including discriminant ,number of soutions and solutions.

Then type:

2 x ^ 2 + 3x - 5 = 0

and click oprtion: solve it!

Then type:

- 4 x ^ 2 - 4x - 1 = 0

and click oprtion: solve it!

To evaluate the discriminant of each equation, we need to use the quadratic formula. The general form of a quadratic equation is ax^2 + bx + c = 0. The discriminant, denoted by Δ (delta), is calculated as follows: Δ = b^2 - 4ac.

Let's evaluate the discriminant for each equation and determine the number and nature of the solutions:

1. Equation: y = 9x^2 - 24x
To calculate the discriminant (Δ), we compare it with the general form (ax^2 + bx + c = 0):
a = 9, b = -24, c = 0
Δ = (-24)^2 - 4 * 9 * 0
Δ = 576 - 0
Δ = 576

The discriminant is positive (Δ > 0). This means that the equation has two distinct real solutions.

2. Equation: y = -2x^2 + 3x - 5
Using the general form:
a = -2, b = 3, c = -5
Δ = (3)^2 - 4 * (-2) * (-5)
Δ = 9 - 40
Δ = -31

The discriminant is negative (Δ < 0). This means that the equation has two complex (imaginary) solutions.

3. Equation: y = -4x^2 - 4x - 1
Using the general form:
a = -4, b = -4, c = -1
Δ = (-4)^2 - 4 * (-4) * (-1)
Δ = 16 - 16
Δ = 0

The discriminant is zero (Δ = 0). This means that the equation has one real solution, also known as a double root.

In summary:

1. Equation: y = 9x^2 - 24x
- Discriminant (Δ) = 576
- Number of solutions: 2
- Solutions are real.

2. Equation: y = -2x^2 + 3x - 5
- Discriminant (Δ) = -31
- Number of solutions: 2
- Solutions are imaginary (complex).

3. Equation: y = -4x^2 - 4x - 1
- Discriminant (Δ) = 0
- Number of solutions: 1
- Solution is real (double root).