Question

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \[ \frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|} \] is equal to____.

Answer 1

Correct Answer: 60

The objective is to determine the smallest possible value for a parameter \( \lambda \) within a quadratic equation. The roots of this equation, denoted by \( \alpha \) and \( \beta \), are restricted to be natural numbers and must satisfy specific divisibility criteria related to \( \lambda \). After finding this minimal \( \lambda \), a designated expression must be evaluated.

Concept Used:

This problem utilizes Vieta's formulas, which establish relationships between the roots and coefficients of a quadratic equation, along with fundamental principles of number theory concerning divisibility.

For a quadratic equation in the standard form \( ax^2 + bx + c = 0 \), with roots \( \alpha \) and \( \beta \), Vieta's formulas are:

• Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)
• Product of roots: \( \alpha \beta = \frac{c}{a} \)

These formulas will be employed to link the roots \( \alpha, \beta \) to \( \lambda \), enabling the application of the given constraints to find the minimum \( \lambda \).

Step-by-Step Solution:

Step 1: Apply Vieta's formulas to the provided equation.

The equation under consideration is \( x^2 - 70x + \lambda = 0 \). Its roots, \( \alpha \) and \( \beta \), are natural numbers (\( \mathbb{N} \)).

According to Vieta's formulas:

\[\alpha + \beta = -(-70) = 70\]\[\alpha \beta = \lambda\]

Step 2: Identify the minimum \( \lambda \) that satisfies the given conditions.

The problem requires \( \lambda \) to be the smallest value such that \( \frac{\lambda}{2} otin \mathbb{N} \) and \( \frac{\lambda}{3} otin \mathbb{N} \). This implies \( \lambda \) must be odd and not divisible by 3.

Given that \( \alpha, \beta \in \mathbb{N} \) and \( \alpha + \beta = 70 \), \( \lambda \) can be expressed in terms of \( \alpha \) as follows:

\[\lambda = \alpha \beta = \alpha(70 - \alpha)\]

To find the minimum value of \( \lambda = 70\alpha - \alpha^2 \), we systematically test values of \( \alpha \) starting from 1.

• For \( \alpha = 1 \), \( \beta = 69 \), \( \lambda = 1 \times 69 = 69 \). Since \( \frac{69}{3} = 23 \in \mathbb{N} \), this is invalid.
• For \( \alpha = 2 \), \( \beta = 68 \), \( \lambda = 2 \times 68 = 136 \). Since \( \frac{136}{2} = 68 \in \mathbb{N} \), this is invalid.
• For \( \alpha = 3 \), \( \beta = 67 \), \( \lambda = 3 \times 67 = 201 \). Since \( \frac{201}{3} = 67 \in \mathbb{N} \), this is invalid.
• For \( \alpha = 4 \), \( \beta = 66 \), \( \lambda = 4 \times 66 = 264 \). This is divisible by 2, so it is invalid.
• For \( \alpha = 5 \), \( \beta = 65 \), \( \lambda = 5 \times 65 = 325 \).
  • Divisibility by 2: 325 is odd, so \( \frac{325}{2} otin \mathbb{N} \). This condition holds.
  • Divisibility by 3: The sum of digits \( 3+2+5=10 \) is not divisible by 3, so \( \frac{325}{3} otin \mathbb{N} \). This condition also holds.

As we are iterating through \( \alpha \) in ascending order, the first \( \lambda \) found that meets all criteria is the minimum. Thus, the minimum \( \lambda \) is 325, corresponding to roots \( \alpha = 5 \) and \( \beta = 65 \).

Step 3: Compute the value of the specified expression.

The expression to be evaluated is:

\[\frac{\left( \sqrt{\alpha - 1} + \sqrt{\beta - 1} \right)(\lambda + 35)}{|\alpha - \beta|}\]

Using \( \alpha = 5 \), \( \beta = 65 \), and \( \lambda = 325 \):

• Numerator term 1: \( \sqrt{\alpha - 1} + \sqrt{\beta - 1} = \sqrt{5 - 1} + \sqrt{65 - 1} = \sqrt{4} + \sqrt{64} = 2 + 8 = 10 \).
• Numerator term 2: \( \lambda + 35 = 325 + 35 = 360 \).
• Denominator: \( |\alpha - \beta| = |5 - 65| = |-60| = 60 \).

Final Computation & Result:

Substituting these values into the expression:

\[\frac{(10)(360)}{60}\]\[= 10 \times \frac{360}{60} = 10 \times 6 = 60\]

The computed value of the expression is 60.