Q.Without solving the equation, find the number of real roots of $2x^2 + 3x + 5 = 0$ using the discriminant.

Compute $D=b^2-4ac$ and check its sign — a negative discriminant means no real root exists.

General form gives $a=2, b=3, c=5$.

Compute the discriminant: $D=b^2-4ac=(3)^2-4(2)(5)=9-40=-31$.

Since $D=-31<0$, by the classification in §6 the equation has no real root — Sridharacharya's formula would require $\sqrt{-31}$, which does not exist within the real numbers $\mathbb{R}$. …

A common mistake is attempting to "force" a numerical answer by ignoring the negative sign under the square root, or by mis-simplifying $\sqrt{-31}$ as if it were a small real correction — within this chapter's scope (real numbers only …