What is true about the solutions of a quadratic equation when the radicand of the quadratic formula is a perfect square?

Answer

No real solutions

Two identical rational solutions

Two different rational solutions

Two irrational solutions

The solution of quadratic formula is represented as

x = - b ± squareroot (b^2 - 4ac) / 2a

Where b^2 - 4ac is known as the radicand.

Lets take an equation; x^2 + 6x + 5 = 0

Where a = 1, b = 6, c = 5

The solution will be x = - 6 ± squareroot (6^2 - 4 (1) (5)) / 2 (1)

Where radicand = 6^2 - 4 (1) (5) = 36 - 20 = 16

Hence radicand is the perfect square of 4. i. e. 4^2 = 16

Now solving the equation further;

x = (-6 ± 4) / 2

x = - 1, x = - 5

Thus for a radicand to be a perfect square, the solution of a quadratic equation will composed of two different rational solutions.