If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p² + q²)x² - 2q(p

Q: Why can't we just use the quadratic formula directly on the first equation?

You absolutely could, but it would be a nightmare of algebra! The coefficients are variables ($p, q, r$), and the discriminant would be a complex expression. In JEE Advanced, whenever you see a quadratic with variables as coefficients, always look for a hidden structure like a perfect square or a symmetric form first.

Q: Wait, why do we reject the root $x = 2$?

Great catch! The problem states that $p, q, r$ do not all have the same sign. If $x = 2$, then $q = 2p$ and $r = 2q = 4p$. If $p$ is positive, $q$ and $r$ are also positive. If $p$ is negative, $q$ and $r$ are also negative. In both cases, they all share the same sign, which violates the constraint. Thus, $x = 2$ is an extraneous solution.

Q: How did you know to group the terms like that? Is there a trick?

It is not a trick, but a pattern recognition skill. When you see $x^2$ and constant terms like $q^2$ and $r^2$ alongside a middle term involving $x$, it is a strong hint to look for perfect squares. By expanding and rearranging, we forced the equation into the form $(px - q)^2 + (qx - r)^2 = 0$, which is a classic technique for solving equations involving real variables.

Question

If for some $p, q, r \in \mathbb{R}$, not all have same sign, one of the roots of the equation $(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0$ is also a root of the equation $x^2 + 2x - 8 = 0$, then $\frac{q^2 + r^2}{p^2}$ is equal to ______.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given the quadratic equation:
$$(p^2 + q^2)x^2 - 2q(p + r)x + q^2 + r^2 = 0$$

At first glance, the presence of variables $p$, $q$, and $r$ appears daunting. However, in JEE Advanced mathematics, complexity is often just simplicity in disguise. Our first task is to peel back these layers through algebraic manipulation.

### Deconstruction: The Art of Grouping

Let us expand the middle term by distributing $-2qx$ across the $(p + r)$ term:
$$(p^2x^2 + q^2x^2) - 2pqx - 2qrx + q^2 + r^2 = 0$$

Now, we rearrange the terms to identify hidden patterns:
$$(p^2x^2 - 2pqx + q^2) + (q^2x^2 - 2qrx + r^2) = 0$$

The monster has been tamed. We recognize these as perfect square trinomials:
$$(px - q)^2 + (qx - r)^2 = 0$$

### The Power of Real Numbers

We are given that $p, q, r$ are real numbers. In the realm of real numbers, the square of any value is non-negative.

If the sum of two non-negative numbers is zero, each individual term must be zero:
$$px - q = 0 \quad 	ext{and} \quad qx - r = 0$$

This leads us to the relations:
$$x = \frac{q}{p} \quad 	ext{and} \quad x = \frac{r}{q}$$

This reveals that $p, q, r$ form a geometric progression with $x$ as the common ratio.

### The Constraint Trap

The problem states that $x$ is also a root of $x^2 + 2x - 8 = 0$. Factoring this quadratic, we obtain:
$$(x + 4)(x - 2) = 0$$

This provides two candidates: $x = 2$ and $x = -4$. We must now satisfy the constraint that $p, q, r$ do not all have the same sign.

If $x = 2$, then $q = 2p$ and $r = 4p$. Regardless of the sign of $p$, all three variables $p, q, r$ will share the same sign. Thus, we must reject $x = 2$.

If $x = -4$, then $q = -4p$ and $r = 16p$. If $p > 0$, then $q < 0$ and $r > 0$. The signs alternate, satisfying our condition.

### The Final Victory

We now evaluate the target expression $\frac{q^2 + r^2}{p^2}$ using $q = -4p$ and $r = 16p$:

$$\frac{(-4p)^2 + (16p)^2}{p^2} = \frac{16p^2 + 256p^2}{p^2}$$

Simplifying the expression:

$$\frac{272p^2}{p^2} = 272$$

We have navigated the complexity and respected the constraints to arrive at the final answer of **272**.

Enter Numerical Value:

Visualized Solution

Analyze the Given Equation

Expand the Middle Term

Regroup the Terms

Identify Perfect Squares

Apply Property of Real Squares

Deduce the Roots

Solve the Second Equation

Check the Sign Constraint (Case 1)

Check the Sign Constraint (Case 2)

Set Up the Final Expression

Calculate the Final Answer

The Sigma Insight: Nature of Roots

Analyzing the Setup

Deconstruction

The Power of Real Numbers

The Constraint Trap

The Final Victory

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