Question

The value of p for which roots of the quadratic equation $x^2 - px + 6 = 0$ are rational, is

• A. $1$
• B. $-5$
• C. $25$
• D. $\sqrt{5}$

Hint:

For multiple-choice questions, directly substitute the options into the equation to see which one leads to simple factorization.
Substituting $p = -5$ gives $x^2 + 5x + 6 = 0$, which instantly factors into $(x+2)(x+3) = 0$!

Answer 1

The Correct Option is B

Step 1: Think in terms of integer roots instead of the discriminant.
In $x^2 - px + 6 = 0$, the leading coefficient is 1, so if the roots are rational they must actually be integers, and both roots must divide 6 exactly (since their product is 6).
Step 2: List the integer factor pairs of 6.
The integer pairs whose product is 6 are $(1,6)$, $(2,3)$, $(-1,-6)$ and $(-2,-3)$.
Step 3: Match the sum of each pair to p.
Since the sum of the roots equals $p$, the possible values of $p$ from these pairs are $7, 5, -7, -5$. Comparing with the given options $1, -5, 25, \sqrt{5}$, only $p = -5$ appears in this list, coming from the roots $-2$ and $-3$.
\[ \boxed{p = -5} \]