In the following, determine whether the given quadratic equation have real roots and if so, find the roots:
3a2x2 + 8abx + 4b2 = 0, a ≠ 0
We have been given, 3a2x2 + 8abx + 4b2 = 0
Now we also know that for an equation ax2 + bx + c = 0, the discriminant is given by the following equation:
D = b2 - 4ac
Now, according to the equation given to us, we have,a = 3a2, b = 8ab and c = 4b2.
Therefore, the discriminant is given as,
D = (8ab)2 - 4(3a2)(4b2)
= 64a2b2 - 48a2b2
Since, in order for a quadratic equation to have real roots, D ≥ 0.Here we find that the equation satisfies this condition, hence it has real roots.
Now, the roots of an equation is given by the following equation,
Therefore, the roots of the equation are given as follows,
Now we solve both cases for the two values of x. So, we have,
Therefore, the roots of the equation are `-(2b)/(3a)` and `(-2b)/a`.