Answer
The two solutions of the equation ${{x}^{2}}+3x-6=0$ are real irrational numbers.
Work Step by Step
$a{{x}^{2}}+bx+c=0$, the radicand ${{b}^{2}}-4ac$ is known as the discriminant.
1. If the value of the discriminant is zero then there is one repeated solution that is rational.
2. When the value of the discriminant is positive, there are two different real rational solutions if the discriminant is a perfect square whereas there are two different real irrational solutions if the discriminant is not a perfect square.
3. If the value of the discriminant is negative then the solutions are imaginary.
${{x}^{2}}+3x-6=0$.
Here,
$\begin{align}
& a=1 \\
& b=3 \\
& c=-6
\end{align}$
So, the value of the discriminant is:
$\begin{align}
& {{b}^{2}}-4ac={{3}^{2}}-4\left( 1 \right)\left( -6 \right) \\
& =9+24 \\
& =33
\end{align}$
The term $33$ is positive and not a perfect square. So, there will be two different real irrational solutions
Therefore, the two solutions of the equation ${{x}^{2}}+3x-6=0$ are real and irrational.
