For the following equation, state the value of the discriminant and then describe the nature of the solutions.
-7x^2+3x-4=0
What is the value of the disriminant?
Does the equation have two imaginary solutions, two real solutions, or one real solution?
First find the discriminant. The formula for it is
D = b^2 - 4ac
D = 3^2 - 4(-7)(-4)
Solve that, and if the solution is positive, there are 2 solutions.
If it's negative, there are imaginary solutions.
If it equals 0, there is one solution
To find the value of the discriminant, we need to use the formula b^2 - 4ac, where the quadratic equation is written in the form ax^2 + bx + c = 0.
In the given equation -7x^2 + 3x - 4 = 0, we can identify that a = -7, b = 3, and c = -4.
The value of the discriminant, D, is calculated as follows:
D = b^2 - 4ac
= (3)^2 - 4(-7)(-4)
= 9 - 112
= -103
The value of the discriminant is -103.
Now, let's determine the nature of the solutions based on the discriminant:
1. If D > 0, the equation has two distinct real solutions.
2. If D = 0, the equation has one real solution (also known as a repeated or double root).
3. If D < 0, the equation has two imaginary solutions.
In this case, since the value of the discriminant is -103 (D < 0), the equation has two imaginary solutions.