If α, β are the roots of the equation x² - (5 + 3^√(log₃ 5) - 5^√(log₅ 3))x + 3(3^((log₃ 5)^(1/3)) -

Q: Why can't we just solve for $\alpha$ and $\beta$ using the quadratic formula? It seems like a standard equation.

That is a trap! If you try to solve for $\alpha$ and $\beta$ directly, you will be stuck in a labyrinth of irrational numbers. The beauty of this problem lies in the structure. We use Vieta's formulas to relate the roots to the coefficients, bypassing the need to find the roots themselves. Always look for the structural shortcut before diving into the quadratic formula.

Q: How do I know when to use the base change property? I always get confused with the reciprocals.

A great rule of thumb: whenever you see a logarithm in the exponent or a fraction involving a log, look for the base change property. If you see $\frac{1}{\log_a b}$, immediately think $\log_b a$. It is the most powerful tool for 'cleaning up' messy logarithmic expressions. If the bases don't match, the base change rule is almost certainly your next move.

Q: Why do the terms cancel out so perfectly? Is it always like this?

In well-constructed JEE problems, yes! The examiners design these expressions to collapse into simple integers. If you find yourself with a massive, unmanageable expression, stop. You have likely missed a simplification step or a property application. Trust the math—it wants to be simple. If it looks impossible, you are likely looking at it the wrong way.

Question

If $\alpha, \beta$ are the roots of the equation $x^2 - \left(5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}}\right)x + 3\left(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1\right) = 0$ then the equation, whose roots are $\alpha + \frac{1}{\beta}$ and $\beta + \frac{1}{\alpha}$, is

Options:

(A) $3x^2 - 20x - 12 = 0$

(B) $3x^2 - 10x - 4 = 0$

(C) $3x^2 - 10x + 2 = 0$

(D) $3x^2 - 20x + 16 = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The "monster" equation presented is a classic example of a JEE Advanced problem designed to test your ability to simplify complex logarithmic expressions before attempting to solve the quadratic.

The equation is given by:
$$x^2 - (5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}})x + 3(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1) = 0$$

Do not panic. This is a puzzle box. The examiners have hidden a simple, elegant truth behind a facade of complexity. Our mission is to strip away the mask.

### Deconstructing the Coefficient of $x$

Let us focus on the coefficient of $x$, which is $5 + 3^{\sqrt{\log_3 5}} - 5^{\sqrt{\log_5 3}}$. We must simplify the term $3^{\sqrt{\log_3 5}}$.

Using the identity $a^{\log_a b} = b$, we rewrite the exponent:
$$3^{\sqrt{\log_3 5}} = (3^{\log_3 5})^{\frac{1}{\sqrt{\log_3 5}}} = 5^{\frac{1}{\sqrt{\log_3 5}}}$$

Applying the base change property, where $\frac{1}{\log_3 5} = \log_5 3$, we find:
$$\frac{1}{\sqrt{\log_3 5}} = \sqrt{\log_5 3}$$

Thus, the term simplifies to $5^{\sqrt{\log_5 3}}$. Substituting this back into the coefficient:
$$5 + 5^{\sqrt{\log_5 3}} - 5^{\sqrt{\log_5 3}} = 5$$

### Simplifying the Constant Term

Now, we address the constant term: $3(3^{(\log_3 5)^{1/3}} - 5^{(\log_5 3)^{2/3}} - 1)$. By applying similar logarithmic manipulations and base change properties, the internal terms simplify significantly.

The expression inside the parenthesis reduces to $-1$. Therefore, the constant term is:
$$3 	imes (-1) = -3$$

Our "monster" equation has now collapsed into the elegant, simple form:
$$x^2 - 5x - 3 = 0$$

### Vieta's Wisdom

We are now on familiar ground. For the equation $x^2 - 5x - 3 = 0$ with roots $\alpha$ and $\beta$, we use Vieta's formulas:
$$\alpha + \beta = 5$$
$$\alpha \beta = -3$$

This is the power of algebraic thinking—we bypass the brute force calculation of the roots entirely.

### The Final Transformation

The problem asks for a new equation with roots $\alpha' = \alpha + \frac{1}{\beta}$ and $\beta' = \beta + \frac{1}{\alpha}$. We simplify these expressions:
$$\alpha' = \frac{\alpha \beta + 1}{\beta} = \frac{-3 + 1}{\beta} = -\frac{2}{\beta}$$
$$\beta' = \frac{\alpha \beta + 1}{\alpha} = \frac{-3 + 1}{\alpha} = -\frac{2}{\alpha}$$

To form the new equation, we calculate the sum $S$ and product $P$ of the new roots:
$$S = \alpha' + \beta' = -2 \left( \frac{1}{\beta} + \frac{1}{\alpha} ight) = -2 \left( \frac{\alpha + \beta}{\alpha \beta} ight) = -2 \left( \frac{5}{-3} ight) = \frac{10}{3}$$
$$P = \alpha' \beta' = \left( -\frac{2}{\beta} ight) \left( -\frac{2}{\alpha} ight) = \frac{4}{\alpha \beta} = \frac{4}{-3} = -\frac{4}{3}$$

The final equation is given by $x^2 - Sx + P = 0$:
$$x^2 - \frac{10}{3}x - \frac{4}{3} = 0$$

Multiplying by $3$ to clear the denominators, we arrive at the final result:
**$3x^2 - 10x - 4 = 0$**

Animated Solution for Mathematics - Quadratic Equations: If $α, β$ are the roots of the equation $x^{2} - (5 + 3^{l o g_{3} 5} - 5^{l o g_{5} 3}) x + 3 (3^{(l o g_{3} 5)^{1/3}} - 5^{(l o g_{5} 3)^{2/3}} - 1) = 0$ then the equation, whose roots are $α + \frac{1}{β}$ and $β + \frac{1}{α}$ , is

Select Answer:

Visualized Solution

Analyze the Given Equation

Simplify the Coefficient of $x$

Apply Logarithmic Identity

Apply Base Change Property

Final Coefficient of $x$

Simplify the Constant Term

Final Constant Term

The Simplified Quadratic Equation

Sum and Product of Roots

Define the New Roots

Substitute Product of Roots

Calculate Sum of New Roots

Evaluate Sum of New Roots

Calculate Product of New Roots

Form the Final Equation

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

Deconstructing the Coefficient of $x$

Simplifying the Constant Term

Vieta's Wisdom

The Final Transformation

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