Answer
If \[{{b}^{2}}-4ac\] is positive and perfect square, then solution is rational and real numbers.
If\[{{b}^{2}}-4ac\]is not a perfect square, then solution is irrational and real numbers.
If \[{{b}^{2}}-4ac\]is negative, then solution is not a real number.
Work Step by Step
The nature of solutions based on the radicand of the quadratic formula, ${{b}^{2}}-4ac$ for the quadratic expression$a{{x}^{2}}+bx+c$.
The radicand of the quadratic formula, ${{b}^{2}}-4ac$, can be used to determine whether
$a{{x}^{2}}+bx+c=0$has solutions that are rational, irrational, or not real numbers.
This can be done by verifying the value of ${{b}^{2}}-4ac$as
If \[{{b}^{2}}-4ac\] is positive and perfect square, then solution is rational and real numbers.
if\[{{b}^{2}}-4ac\]is not a perfect square, then solution is irrational and real numbers.
If \[{{b}^{2}}-4ac\]is negative, then solution is not a real number.
So, it is possible to determine the nature of solutions by application of above principles
without actual solving.
