Question

For what positive integral values of x is x^2 - 5bx + 3b = 0 so that the two roots are real and distinct?

a) b > 0
b) b < 0
c) b = 0
d) b ≠ 0

Answer 1

For the quadratic equation x^2 - 5bx + 3b = 0 to have real and distinct roots, the values of b should be greater than 0 or less than 12/25.

Step-by-step explanation:

To find the values of b for which the quadratic equation x^2 - 5bx + 3b = 0 has real and distinct roots, we need to apply the conditions for discriminant. The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.

For the roots to be real and distinct, the discriminant must be greater than 0. In this case, the discriminant is (-5b)^2 - 4(1)(3b) = 25b^2 - 12b.

To find the values of b, we set this expression greater than 0 and solve for b:

25b^2 - 12b > 0.

This expression is factorable as b(25b - 12) > 0. This inequality holds true for b > 0 and b < 12/25.