Use the discriminant to determine the number of real roots the equation has.
3x2 – 5x + 1 =0
A. One real root (a double root)
B. Two distinct real roots
C. Three real roots
D. None (two imaginary roots)
To determine the number of real roots of the equation 3x^2 - 5x + 1 = 0, we can use the discriminant formula. The discriminant is calculated using the coefficients of the quadratic equation and is given by the formula:
Discriminant = b^2 - 4ac
In this equation, a = 3, b = -5, and c = 1. Plugging these values into the formula:
Discriminant = (-5)^2 - 4(3)(1) = 25 - 12 = 13
Now, let's analyze the value of the discriminant:
If the discriminant is positive (greater than 0), then the equation has two distinct real roots. (Option B)
If the discriminant is zero, then the equation has one real root (a double root). (Option A)
If the discriminant is negative (less than 0), then the equation has no real roots; it has two imaginary roots. (Option D)
In our case, the discriminant is positive (13), so the equation has two distinct real roots. Therefore, the correct answer is option B: Two distinct real roots.