erisedk said:The Attempt at a Solution
1-4(a)(c-a)<0
4ac > 4a^2 + 1
PeroK said:Okay, so far. What sort of inequality is that in relation to a?
... although, I don't see how you get that answer and c > 1 would make more sense. Are you sure you've got the question right?
erisedk said:Both roots are complex is the same as both roots are imaginary, I think.
The question is correct, at least should be since it's a question from a test I took yesterday.
I don't have any idea what sort of inequality that is in relation to a.
erisedk said:Wait, why is the condition 3a<2 not relevant?
Imaginary roots are solutions to a polynomial equation that involve the imaginary unit, which is denoted by i. The imaginary unit is defined as the square root of -1. Imaginary roots are necessary to solve certain equations that cannot be solved using only real numbers.
To find imaginary roots, you must first express the polynomial equation in the form ax^2 + bx + c = 0. Then, use the quadratic formula x = (-b ± √(b^2-4ac)) / 2a to solve for the roots. If the discriminant (b^2-4ac) is negative, the roots will be imaginary.
Vieta's formula is a theorem that relates the coefficients of a polynomial to the sum and product of its roots. For a quadratic equation ax^2 + bx + c = 0, the sum of the roots is equal to -b/a and the product of the roots is equal to c/a.
In this equation, the coefficients are 3, 0, and -2. Using Vieta's formula, we can see that the sum of the roots is equal to 0/3 = 0 and the product of the roots is equal to -2/3. Therefore, the equation has no real roots, but it may have imaginary roots depending on the value of c.
Imaginary roots and Vieta's formula are important in mathematics because they allow us to solve more complex equations and understand the relationships between the coefficients and roots of a polynomial. They are also used in various fields such as engineering, physics, and computer science to model and solve real-world problems.