JEE Main 2021 (20 July Shift 1)

LEVELJEE Main

Animated Solution for Mathematics - Quadratic Equations: If $α$ and $β$ are the distinct roots of the equation $x^{2} + (3)^{1/4} x + 3^{1/2} = 0$ , then the value of $α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$ is equal to :

Select Answer:

Visualized Solution

Understanding the Root Equation

Given equation: $x^{2} + 3^{\frac{1}{4}} x + 3^{\frac{1}{2}} = 0$
Roots are $α$ and $β$ .
Objective: Find the value of $α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$ .

Isolating the Fractional Power

Rearrange the equation to isolate the $x$ term:
$(x^{2} + 3^{\frac{1}{2}}) = - 3^{\frac{1}{4}} x$

First Squaring Operation

Square both sides:
$(x^{2} + 3^{\frac{1}{2}})^{2} = (- 3^{\frac{1}{4}} x)^{2}$

Expanding the Squared Terms

Expand the left hand side using $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ :
$(x^{2})^{2} + 2 (x^{2}) (3^{\frac{1}{2}}) + (3^{\frac{1}{2}})^{2} = 3^{\frac{1}{2}} x^{2}$
Simplifying: $x^{4} + 2 \cdot 3^{\frac{1}{2}} x^{2} + 3 = 3^{\frac{1}{2}} x^{2}$

Simplifying the Equation

Subtract $3^{\frac{1}{2}} x^{2}$ from both sides:
$x^{4} + 3^{\frac{1}{2}} x^{2} + 3 = 0$

Preparing for the Second Square

Isolate the radical term again:
$x^{4} + 3 = - 3^{\frac{1}{2}} x^{2}$

Second Squaring Operation

Square both sides again:
$(x^{4} + 3)^{2} = (- 3^{\frac{1}{2}} x^{2})^{2}$

Expanding and Final Polynomial

Expand the LHS:
$(x^{4})^{2} + 2 (x^{4}) (3) + 3^{2} = 3 x^{4}$
Simplify: $x^{8} + 6 x^{4} + 9 = 3 x^{4}$
Final simplified form: $x^{8} + 3 x^{4} + 9 = 0$

The Clever Algebraic Identity

Multiply the equation by $(x^{4} - 3)$ :
$(x^{4} - 3) (x^{8} + 3 x^{4} + 9) = 0$
Using the identity $(a - b) (a^{2} + ab + b^{2}) = a^{3} - b^{3}$ :
$(x^{4})^{3} - 3^{3} = 0$
Therefore, $x^{12} - 27 = 0 \Rightarrow x^{12} = 27$

Finding Alpha to the Power 96

Since $α$ and $β$ are roots: $α^{12} = 27$ and $β^{12} = 27$ .
Calculate $α^{96}$ :
$α^{96} = (α^{12})^{8} = (27)^{8}$
Substitute $27 = 3^{3}$ :
$α^{96} = (3^{3})^{8} = 3^{24}$
Similarly, $β^{96} = 3^{24}$

Evaluating the Target Expression

Substitute the values into the expression:
$E = α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$
Substitute $α^{12} = 27$ and $α^{96} = 3^{24}$ :
$E = 3^{24} (27 - 1) + 3^{24} (27 - 1)$

Final Arithmetic and Conclusion

Simplify the terms:
$E = 3^{24} (26) + 3^{24} (26)$
Factor out $3^{24}$ :
$E = 3^{24} (26 + 26)$
Final result: $E = 52 \times 3^{24}$
Correct Option: (3)

00:00 / 00:00

The Sigma Insight: Solution of Quadratic Equations

Analyzing the Setup

We begin with the quadratic equation:

x^{2} + 3^{1/4} x + 3^{1/2} = 0

The roots of this equation are

α

and

β

. Our objective is to evaluate the expression:

E = α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)

The presence of high powers like

96

and

12

suggests that the roots satisfy a higher-order polynomial equation. We must eliminate the radicals to uncover this structure.

The Art of Isolation

First, we isolate the term containing the

3^{1/4}

radical:

x^{2} + 3^{1/2} = - 3^{1/4} x

Squaring both sides yields:

(x^{2} + 3^{1/2})^{2} = (- 3^{1/4} x)^{2}

x^{4} + 2 \cdot 3^{1/2} x^{2} + 3 = 3^{1/2} x^{2}

Rearranging the terms, we obtain:

x^{4} + 3^{1/2} x^{2} + 3 = 0

The Second Square and the Hidden Identity

To eliminate the remaining radical, we isolate

3^{1/2} x^{2}

x^{4} + 3 = - 3^{1/2} x^{2}

Squaring both sides once more:

(x^{4} + 3)^{2} = (- 3^{1/2} x^{2})^{2}

x^{8} + 6 x^{4} + 9 = 3 x^{4}

Subtracting

3 x^{4}

from both sides results in the clean polynomial:

x^{8} + 3 x^{4} + 9 = 0

Recognizing this as the form

a^{2} + ab + b^{2}

where

a = x^{4}

and

b = 3

, we multiply by

(x^{4} - 3)

(x^{4} - 3) (x^{8} + 3 x^{4} + 9) = (x^{4})^{3} - 3^{3}

x^{12} - 27 = 0 \Rightarrow x^{12} = 27

Final Calculation

Since

α

and

β

are roots of the original equation, they must satisfy

x^{12} = 27

. Consequently,

α^{12} = 27

and

β^{12} = 27

We now substitute these values into our target expression

E

E = α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)

E = (α^{12})^{8} (27 - 1) + (β^{12})^{8} (27 - 1)

Substituting

27 = 3^{3}

E = (3^{3})^{8} (26) + (3^{3})^{8} (26)

E = 3^{24} (26) + 3^{24} (26)

Combining the terms, we reach the final result: $E = 52 \times 3^{24}$

Common Traps & FAQs

Wait, why can't we just solve for

x

using the quadratic formula?

Using the quadratic formula here would lead you into a nightmare of nested radicals and complex numbers. The problem is designed to test your ability to manipulate algebraic structures rather than brute-force calculation. Since the target expression is symmetric in

α

and

β

, we only need to find a relationship that both roots satisfy, not the roots themselves.

How did you know to multiply by

(x^{4} - 3)

That is the 'Aha!' moment! When you see a polynomial of the form

x^{8} + 3 x^{4} + 9

, it perfectly matches the expansion of

(x^{4})^{2} + (x^{4}) (3) + 3^{2}

. This is the second factor of the difference of cubes identity

a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2})

. Multiplying by

(a - b)

, which is

(x^{4} - 3)

, collapses the entire expression into a simple power of

x

Is it always valid to square both sides of an equation?

Squaring both sides is a powerful tool, but it can introduce extraneous roots. However, in this context, we are not solving for

x

to find its specific value; we are deriving a polynomial equation that the roots

α

and

β

must satisfy. As long as the original roots are preserved in the new polynomial, the logic holds perfectly.

Animated Solution for Mathematics - Quadratic Equations: If $α$ and $β$ are the distinct roots of the equation $x^{2} + (3)^{1/4} x + 3^{1/2} = 0$ , then the value of $α^{96} (α^{12} - 1) + β^{96} (β^{12} - 1)$ is equal to :

Select Answer:

Visualized Solution

Understanding the Root Equation

Isolating the Fractional Power

First Squaring Operation

Expanding the Squared Terms

Simplifying the Equation

Preparing for the Second Square

Second Squaring Operation

Expanding and Final Polynomial

The Clever Algebraic Identity

Finding Alpha to the Power 96

Evaluating the Target Expression

Final Arithmetic and Conclusion

The Sigma Insight: Solution of Quadratic Equations

Analyzing the Setup

The Art of Isolation

The Second Square and the Hidden Identity

Final Calculation

Similar Questions

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Let $α_{1}, α_{2}, \dots, α_{7}$ be the roots of the equation $x^{7} + 3 x^{5} - 13 x^{3} - 15 x = 0$ and $∣ α_{1} ∣ \geq ∣ α_{2} ∣ \geq \dots \geq ∣ α_{7} ∣$ . Then $α_{1} α_{2} - α_{3} α_{4} + α_{5} α_{6}$ is equal to ______.

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The sum of all real values of $x$ satisfying the equation $(x^{2} - 5 x + 5)^{x^{2} + 4 x - 60} = 1$ is :

Select Answer:

Visualized Solution

Understanding the Root Equation

Isolating the Fractional Power

First Squaring Operation

Expanding the Squared Terms

Simplifying the Equation

Preparing for the Second Square

Second Squaring Operation

Expanding and Final Polynomial

The Clever Algebraic Identity

Finding Alpha to the Power 96

Evaluating the Target Expression

Final Arithmetic and Conclusion

The Sigma Insight: Solution of Quadratic Equations

Analyzing the Setup

The Art of Isolation

The Second Square and the Hidden Identity

Final Calculation

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