Explanation
Let's assume that we have a quadratic equation of the form:
[tex]ax^2+bx+c=0[/tex]
Where a, b and c are all real numbers. The solution(s) to this equation are given by the quadratic solving formula:
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Where the term inside the square root is known as the discriminant of the equation. Depending on its value we have three cases:
- The discriminant is greater than 0 and the equation has two real solutions.
- The discriminant is equal to 0 and the equation has one real solution.
- The discriminant is smaller than 0 and the equation has no real solutions i.e. two nonreal solutions.
In this case we have the equation:
[tex]6x^2-4x+3=0[/tex]
So a=6, b=-4 and c=3. Then the discriminant of this equation is:
[tex]b^2-4ac=(-4)^2-4\cdot6\cdot3=16-72=-56[/tex]
So the discriminant is smaller than 0 which means that this equation has two nonreal solutions.
Answer
So the first answer is:
Discriminant: -56
And the second answer is the third option, two distinct nonreal solutions.
