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:

To quickly solve quadratic equation root-nature problems, try substituting the options directly into the equation.
With $p = -5$, the equation factors cleanly into $(x+2)(x+3) = 0$, giving integer roots immediately!

Answer 1

The Correct Option is B

Step 1: Think in terms of integer root pairs instead of the discriminant.
In $x^2 - px + 6 = 0$, the product of the roots is $6$ and the sum of the roots is $p$. If the roots are rational (and the equation has integer coefficients), they are most naturally a pair of integers multiplying to $6$.
Step 2: List integer factor pairs of 6 and their sums.
$(1,6) \to$ sum $7$; $(2,3) \to$ sum $5$; $(-1,-6) \to$ sum $-7$; $(-2,-3) \to$ sum $-5$.
Step 3: Match a sum to the given options.
Among $\{7,5,-7,-5\}$, only $-5$ appears in the answer choices. So $p = -5$, coming from roots $-2$ and $-3$ (check: $(-2)+(-3)=-5=p$, $(-2)(-3)=6$ ✓).
\[ \boxed{p = -5} \]