Question

For what positive integers c, with c < 100, does the following quadratic have rational roots? 3x^2 + 20x + c

Answer 1

There are rational roots for $c = \frac{100}{3}$.

How to find if second order polynomials are real and rational

For second order polynomials of the form $a\cdot x^{2} + b\cdot x + c = 0$, where $a, b,c \in \mathbb{R}$. roots are real if $b^{2}-4\cdot a\cdot c \ge 0$. In addition, roots are rational if $b^{2}-4\cdot a \cdot c = 0$.

If we know that $a = 3$, $b = 20$ and $c < 100$, then the discriminant must be $400-12\cdot c \ge 0$. We must consider the following case:

Discriminant is 0

$400 - 12\cdot c = 0$

$c = \frac{100}{3}$

There are rational roots for $c = \frac{100}{3}$. $\blacksquare$

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