If a, b, c are in G.P., then the equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common roo

Q: Wait, why do we reject the negative root here?

We actually don't reject it! In the equation $(\sqrt{a}x + \sqrt{c})^2 = 0$, the root is $x = -\sqrt{\frac{c}{a}}$. Because the expression is a perfect square, this is a repeated root (the parabola touches the x-axis at exactly one point). Since the problem states the two equations have a common root, this repeated root must be that common root.

Q: Why can't we just use the discriminant condition for a common root?

You absolutely could! You could set the discriminant of both equations to zero or use the condition for a common root involving coefficients. However, that path is algebraically tedious. The 'perfect square' observation is a beautiful shortcut that reveals the underlying symmetry of the problem much faster.

Q: How did you know to divide by $c$ in the final step?

Great question! Always look at the target expression. We are asked to prove that $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P. By dividing the equation by $c$, we naturally generate the terms $\frac{d}{a}$ and $\frac{f}{c}$. It is a strategic move to align our algebraic expression with the form required for an Arithmetic Progression.

Question

If $a, b, c$ are in G.P., then the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root if $d/a, e/b, f/c$ are in

Options:

(A) A.P.

(B) G.P.

(C) H.P.

(D) none of these

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the mathematical landscape. Today, we are going to unravel a problem that, at first glance, looks like a standard exercise in quadratic equations, but beneath the surface, it hides a beautiful, elegant symmetry.

We are given two quadratic equations, $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$, and a tantalizing clue: $a, b, c$ are in Geometric Progression (G.P.). Our mission is to find the relationship between the ratios $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$.

### The G.P. Secret

Let us start by honoring the G.P. condition. If $a, b, c$ are in G.P., the fundamental property is that the square of the middle term equals the product of the extremes: $b^2 = ac$.

This implies $b = \sqrt{ac}$. This allows us to rewrite the first equation, $ax^2 + 2bx + c = 0$, as:

$$ax^2 + 2\sqrt{ac}x + c = 0$$

### The Perfect Square Revelation

Look closely at the equation above. It is a perfect square! We can rewrite it as:

$$(\sqrt{a}x)^2 + 2(\sqrt{a}x)(\sqrt{c}) + (\sqrt{c})^2 = 0$$

This simplifies to:

$$(\sqrt{a}x + \sqrt{c})^2 = 0$$

This is the moment of clarity. The equation has a repeated root, $x = -\sqrt{\frac{c}{a}}$. The parabola defined by the first equation doesn't just cross the x-axis; it kisses it at this single point. This is our common root.

### The Bridge Between Equations

Since the two equations share a common root, and the first equation has only one unique root, that root MUST be the common one. We substitute $x = -\sqrt{\frac{c}{a}}$ into the second equation:

$$d\left(-\sqrt{\frac{c}{a}}\right)^2 + 2e\left(-\sqrt{\frac{c}{a}}\right) + f = 0$$

Squaring the root gives us:

$$d\left(\frac{c}{a}\right) - 2e\sqrt{\frac{c}{a}} + f = 0$$

We rearrange this to isolate the term with the square root:

$$\frac{dc}{a} + f = 2e\sqrt{\frac{c}{a}}$$

### The Final Transformation

To reach the form involving $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$, we divide the entire equation by $c$:

$$\frac{d}{a} + \frac{f}{c} = \frac{2e}{c}\sqrt{\frac{c}{a}}$$

Simplifying the right side, we get:

$$\frac{d}{a} + \frac{f}{c} = \frac{2e}{\sqrt{ac}}$$

Recalling our initial G.P. condition, $b = \sqrt{ac}$, we substitute it back to obtain:

$$\frac{d}{a} + \frac{f}{c} = \frac{2e}{b}$$

This is the classic definition of an Arithmetic Progression: the sum of the first and third terms is twice the middle term. Thus, **$\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in A.P.** We have arrived at the destination, and the path was paved with pure, logical elegance.

Animated Solution for Mathematics - Quadratic Equations: If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

Select Answer:

Visualized Solution

Analyze the G.P. Condition

Substitute $b$ in the First Equation

Identify the Perfect Square

Solve for the Common Root $x$

Substitute the Root into the Second Equation

Simplify the Terms

Divide by $c$ and Use G.P. Relation

Conclusion: Identify the A.P.

The Sigma Insight: Common Roots

Analyzing the Setup

The G.P

The Perfect Square Revelation

The Bridge Between Equations

The Final Transformation

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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 a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in