If one root of the equation x² + px + 12 = 0 is 4, while the equation x² + px + q = 0 has equal root

Q: Wait, why do we substitute $x = 4$ into the first equation? Can't we just use the sum of roots formula?

That is a great question! You could use the sum of roots formula if you knew the other root, but you don't. The definition of a root is that it satisfies the equation. By substituting $x = 4$, you are using the most direct path to find the unknown coefficient $p$ without needing any extra information.

Q: What does 'equal roots' actually mean on a graph?

Think of a parabola as a curve that usually crosses the $x$-axis at two points. When a quadratic has 'equal roots', the parabola is shifted such that its vertex just barely touches the $x$-axis. It doesn't cross it; it kisses it! This is why the discriminant must be zero—it represents the transition point between having two real roots and no real roots.

Q: Is it possible for $p$ to have two values? Why did we only get one?

In this specific problem, $p$ is uniquely determined by the fact that $x=4$ is a root of the first equation. Since the equation is linear in terms of $p$ once $x$ is substituted, there is only one possible value for $p$. Don't worry—if the problem had been structured differently, you might have had to consider multiple cases, but here, the math is straightforward and singular.

Question

If one root of the equation $x^2 + px + 12 = 0$ is 4, while the equation $x^2 + px + q = 0$ has equal roots, then the value of 'q' is

Options:

(A) 4

(B) 12

(C) 3

(D) 49/4

Sigma Percentile · Accepted Answer

### The Beauty of Quadratic Equations: A Journey of Discovery

Welcome, future IITians! Today, we are going to unravel a classic problem that sits at the heart of JEE Advanced preparation. Quadratic equations are not just algebraic expressions; they are the language of parabolas, trajectories, and the very geometry of our physical world.

Let's dive into this problem with curiosity and precision.

### Phase 1: The First Key

We are presented with two quadratic equations:
1. $x^2 + px + 12 = 0$
2. $x^2 + px + q = 0$

We are told that $x = 4$ is a root of the first equation. Now, I want you to pause and think: what does it mean for a number to be a root? It is the ultimate key.

If you plug $x = 4$ into the equation, the entire expression must collapse to zero. It is a statement of truth. So, let's perform this substitution:

$$ (4)^2 + p(4) + 12 = 0 $$

Look at that! We have transformed a quadratic equation into a simple linear equation in terms of $p$. Let's simplify it step-by-step:

$$ 16 + 4p + 12 = 0 $$

$$ 28 + 4p = 0 $$

By subtracting $28$ from both sides, we get $4p = -28$. Dividing by $4$, we find our first victory: **$p = -7$**. This coefficient is the bridge between our two equations.

### Phase 2: The Geometric Insight

Now that we have $p = -7$, our second equation becomes $x^2 - 7x + q = 0$. The problem tells us something fascinating: this equation has 'equal roots'.

Imagine the graph of a quadratic function $f(x) = x^2 - 7x + q$. Usually, a parabola cuts through the $x$-axis at two distinct points. But when the roots are equal, the parabola is perfectly positioned so that its vertex just touches the $x$-axis.

Algebraically, this is the moment where the discriminant, $\Delta = b^2 - 4ac$, is exactly zero. It is the boundary between having two real roots and having no real roots at all.

### Phase 3: The Final Calculation

Let's apply this condition. For our equation $x^2 - 7x + q = 0$, the coefficients are $a = 1$, $b = -7$, and $c = q$. The discriminant formula is our trusted tool:

$$ \Delta = b^2 - 4ac = 0 $$

Substituting our values, we get:

$$ (-7)^2 - 4(1)(q) = 0 $$

$$ 49 - 4q = 0 $$

This is the final stretch. We transpose the $4q$ to the other side:

$$ 4q = 49 $$

$$ q = \frac{49}{4} $$

And there it is! The value of $q$ is **$\frac{49}{4}$**.

### Conclusion

Take a moment to appreciate the elegance of what we just did. We used the definition of a root to unlock the first equation, and then we used the geometric property of equal roots to solve the second.

This is the essence of JEE mathematics: connecting algebraic manipulation with geometric intuition. Keep this mindset, and you will find that no problem is too complex to solve. Keep practicing, stay curious, and keep pushing toward your goal!

Animated Solution for Mathematics - Quadratic Equations: If one root of the equation $x^{2} + p x + 12 = 0$ is 4, while the equation $x^{2} + p x + q = 0$ has equal roots, then the value of 'q' is

Select Answer:

Visualized Solution

Visualizing the Problem & First Equation

The Definition of a Root

Substituting $x = 4$

Solving for Coefficient $p$

Introducing the Second Equation

Understanding the 'Equal Roots' Condition

Setting up the Discriminant

Solving for $q$

Summary & Key Takeaways

The Sigma Insight: Nature of Roots

The Beauty of Quadratic Equations

Phase 1

Phase 2

Phase 3

Conclusion

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the equation x2+px+12=0 is 4, while the equation x2+px+q=0 has equal roots, then the value of 'q' is