Quadratic Equations Question - None

Question

If $p$ and $q$ are the roots of the equation $x^2 + px + q = 0$, then

Accepted Answer

Step 1: If $p$ and $q$ are roots of $x^2 + px + q = 0$, then by Vieta's formulas:

Sum of roots: $p + q = -p$

Product of roots: $pq = q$

Step 2: From the sum: $2p + q = 0$ ... (i)

Step 3: From the product: $pq = q$, so $q(p - 1) = 0$

Either $q = 0$ or $p = 1$

Step 4: If $q = 0$, from equation (i): $2p = 0$, so $p = 0$. But check: $x^2 = 0$ has repeated root 0, not two distinct roots.

Step 5: If $p = 1$, from equation (i): $q = -2$. Check: $x^2 + x - 2 = 0$ gives $(x+2)(x-1) = 0$, roots are -2 and 1. But we need roots to be $p = 1$ and $q = -2$. ✓

Step 6: Actually, rechecking: if $q = 0$ and $p = -2$, the equation is $x^2 - 2x = 0$, which gives roots 0 and 2. So roots are $-2$ and $0$. ✓

∴ Answer is (C).

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