If p and q are the roots of the equation x² + px + q = 0, then (A) p = 1, q = -2 (B) p = 0, q = 1 (C

Q: Wait, why can't I just divide by $q$ in the equation $pq = q$?

That is a classic trap! If you divide by $q$, you implicitly assume $q \neq 0$. However, $q$ could be zero. Always factor instead: $pq - q = 0$ becomes $q(p - 1) = 0$. This reveals two distinct possibilities: $q = 0$ or $p = 1$.

Q: What if $p=0$ and $q=0$? Why isn't that the answer?

If $p=0$ and $q=0$, the equation becomes $x^2 = 0$, which has roots $0$ and $0$. This is mathematically valid, but if you look at the provided options, $(0, 0)$ is not listed. In JEE problems, always check your derived solutions against the given options.

Q: How do I know which case to pick between $q=0$ and $p=1$?

You test them! If $q=0$, the first equation $2p + q = 0$ forces $p=0$. If $p=1$, the first equation $2p + q = 0$ forces $q = -2$. Since $(0, 0)$ isn't an option, $(1, -2)$ is the only valid path forward.

Question

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

Options:

(A) $p = 1, q = -2$

(B) $p = 0, q = 1$

(C) $p = -2, q = 0$

(D) $p = -2, q = 1$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are staring at the equation $x^2 + px + q = 0$. It looks simple, almost unassuming.

But there is a hidden depth here: the coefficients $p$ and $q$ are not just random numbers; they are the roots of the very equation they define. This is a beautiful, self-referential puzzle.

To solve it, we must step into the world of Vieta's formulas, the bridge between the roots of a polynomial and its coefficients.

### The First Bridge: Sum of Roots

Let us recall the wisdom of Vieta. For any quadratic equation $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$.

In our specific case, $a = 1$, $b = p$, and $c = q$. The roots are $p$ and $q$.

Therefore, the sum of the roots is $p + q$, and according to the formula, this must equal $-\frac{p}{1}$, which is simply $-p$.

So, we arrive at our first vital relationship:
$$p + q = -p$$

With a quick algebraic shuffle, moving that $-p$ to the left, we get:
$$2p + q = 0$$

This is our first anchor point.

### The Second Bridge: Product of Roots

Now, we need more information. Vieta tells us that the product of the roots is $\frac{c}{a}$.

For our equation, the product of the roots $p$ and $q$ is $p 	imes q$, and the constant term $c$ is $q$. Thus, we have the equation:
$$pq = q$$

Here is where many students stumble. The instinct is to divide both sides by $q$, but that is a dangerous move!

If $q$ happens to be zero, you have just divided by zero, which is undefined. Instead, let us bring $q$ to the left side:
$$pq - q = 0$$

Factoring out the common $q$, we get:
$$q(p - 1) = 0$$

This elegant factorization reveals two distinct cases: either $q = 0$ or $p = 1$.

### The Resolution

Let us test these cases. Case 1: If $q = 0$, we substitute this into our first anchor equation, $2p + q = 0$.

This gives us $2p + 0 = 0$, which means $p = 0$. So, one potential solution is $p = 0, q = 0$.

However, we must move to Case 2: $p = 1$. Substituting $p = 1$ into our first equation, $2p + q = 0$, we get:
$$2(1) + q = 0$$

This simplifies to $2 + q = 0$. Solving for $q$, we find $q = -2$.

We have arrived at our destination: **$p = 1$ and $q = -2$**.

If you plug these values back into the original equation, you get $x^2 + x - 2 = 0$. The roots of this equation are indeed $1$ and $-2$.

The math is consistent, the logic is sound, and the geometry of the parabola perfectly aligns with our algebraic findings. You have successfully navigated the trap and found the truth.

Animated Solution for Mathematics - Quadratic Equations: If $p$ and $q$ are the roots of the equation $x^{2} + p x + q = 0$ , then

Select Answer:

Visualized Solution

The Quadratic Equation

Sum of Roots Property

Applying Sum of Roots

Simplifying the First Equation

Product of Roots Property

Applying Product of Roots

Solving the Product Equation

Evaluating the Cases

Finding the Value of $q$

Final Solution and Visualization

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The First Bridge

The Second Bridge

The Resolution

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