Question

If \(\alpha, \beta\) are the roots of the equation \(x^{2}-px+q=0\) and \(\alpha>0\), \(\beta>0\), then \[ \alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=(p+6\sqrt{q}+4q^{\frac{1}{4}}\sqrt{p+2\sqrt{q}})^{\kappa}, \] where \(\kappa\) is:

• A. $\frac{3}{2}$
• B. $2$
• C. $\frac{1}{3}$
• D. $\frac{1}{4}$

Hint:

When resolving complicated radical identities containing unknown exponents, try substituting simple numbers! If we let $\alpha = 1$ and $\beta = 1$, then $p = 2$ and $q = 1$. The left side becomes $1+1=2$, while the inside right expression simplifies to $2 + 6(1) + 4(1)\sqrt{2+2} = 8 + 4(2) = 16$. Solving $2 = (16)^K$ immediately yields $K = \frac{1}{4}$!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires the application of the relations between the roots and coefficients of a quadratic equation. For the quadratic \( ax^2 + bx + c = 0 \), the sum of roots is \( -b/a \) and the product is \( c/a \). We need to manipulate these relations algebraically to find an expression for the sum of the fourth roots of the roots.
Step 2: Key Formula or Approach:
1. Sum of roots: \( \alpha + \beta = p \).
2. Product of roots: \( \alpha\beta = q \).
3. We will build the expression starting from \( \sqrt{\alpha} + \sqrt{\beta} \) and then move to \( \alpha^{1/4} + \beta^{1/4} \) by repeated squaring or reverse substitution.
Step 3: Detailed Explanation:
First, let's find the value of \( \sqrt{\alpha} + \sqrt{\beta} \).
We know that \( (\sqrt{\alpha} + \sqrt{\beta})^2 = \alpha + \beta + 2\sqrt{\alpha\beta} \).
Substituting the values of \( p \) and \( q \):
\[ (\sqrt{\alpha} + \sqrt{\beta})^2 = p + 2\sqrt{q} \implies \sqrt{\alpha} + \sqrt{\beta} = \sqrt{p + 2\sqrt{q}} \]
Next, let \( Y = \alpha^{1/4} + \beta^{1/4} \).
Square \( Y \):
\[ Y^2 = (\alpha^{1/4} + \beta^{1/4})^2 = \sqrt{\alpha} + \sqrt{\beta} + 2(\alpha\beta)^{1/4} \]
Substituting the result for \( \sqrt{\alpha} + \sqrt{\beta} \):
\[ Y^2 = \sqrt{p + 2\sqrt{q}} + 2q^{1/4} \]
Now, let's look at the expression provided in the question: \( Z = p + 6\sqrt{q} + 4q^{1/4} \sqrt{p + 2\sqrt{q}} \).
Let's see if we can relate \( Z \) to \( Y \). Consider squaring \( Y^2 \), which is \( Y^4 \):
\[ Y^4 = (\sqrt{p + 2\sqrt{q}} + 2q^{1/4})^2 \]
Expand using \( (a+b)^2 = a^2 + b^2 + 2ab \):
\[ Y^4 = (\sqrt{p + 2\sqrt{q}})^2 + (2q^{1/4})^2 + 2(\sqrt{p + 2\sqrt{q}})(2q^{1/4}) \]
\[ Y^4 = (p + 2\sqrt{q}) + 4\sqrt{q} + 4q^{1/4} \sqrt{p + 2\sqrt{q}} \]
\[ Y^4 = p + 6\sqrt{q} + 4q^{1/4} \sqrt{p + 2\sqrt{q}} \]
Notice that the expression inside the parenthesis in the question is exactly \( Y^4 \).
So the question states: \( Y = (Y^4)^{\kappa} \).
For this to hold true:
\[ Y = Y^{4\kappa} \implies 4\kappa = 1 \implies \kappa = \frac{1}{4} \]
Step 4: Final Answer:
Through algebraic manipulation and squaring the sum of the fourth roots twice, we found that the given complex expression is actually the fourth power of \( \alpha^{1/4} + \beta^{1/4} \). Thus, \( K \) must be \( 1/4 \).