Analyzing Quadratic Roots Difference Condition

The question requires finding the sum of all possible integer values for $\lambda$ in the equation $12x^2 - 20x + 3\lambda = 0$, subject to the condition $\frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2}$, where $\alpha$ and $\beta$ are the roots.

Calculating Root Properties

From the quadratic equation $12x^2 - 20x + 3\lambda = 0$, we identify coefficients $a=12$, $b=-20$, $c=3\lambda$. The sum of roots is $\alpha + \beta = -\frac{b}{a} = -\frac{-20}{12} = \frac{5}{3}$. The product of roots is $\alpha \beta = \frac{c}{a} = \frac{3\lambda}{12} = \frac{\lambda}{4}$.

Deriving Root Difference Formula

The square of the difference between roots is $(\beta - \alpha)^2 = (\alpha + \beta)^2 - 4\alpha\beta$. Substituting the values:

$ (\beta - \alpha)^2 = \left(\frac{5}{3}\right)^2 - 4\left(\frac{\lambda}{4}\right) = \frac{25}{9} - \lambda $

Thus, the absolute difference is $|\beta - \alpha| = \sqrt{\frac{25}{9} - \lambda}$. For real roots, we must have $\frac{25}{9} - \lambda \ge 0$, which implies $\lambda \le \frac{25}{9} \approx 2.777...$.

Applying the Inequality Condition

The given condition is $\frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2}$. Squaring all parts of the inequality:

$ \frac{1}{4} \leq |\beta - \alpha|^2 \leq \frac{9}{4} $

Substituting the expression for $(\beta - \alpha)^2$:

$ \frac{1}{4} \leq \frac{25}{9} - \lambda \leq \frac{9}{4} $

Solving for $\lambda$ (Real Roots)

Rearranging the inequality to solve for $\lambda$:

$ \frac{1}{4} - \frac{25}{9} \leq -\lambda \leq \frac{9}{4} - \frac{25}{9} $

$ \frac{9 - 100}{36} \leq -\lambda \leq \frac{81 - 100}{36} $

$ -\frac{91}{36} \leq -\lambda \leq -\frac{19}{36} $

Multiplying by -1 reverses the inequality signs:

$ \frac{19}{36} \leq \lambda \leq \frac{91}{36} $

Approximately, $0.527... \leq \lambda \leq 2.527...$. Since $\lambda$ must be an integer and satisfy $\lambda \le 2.777...$ (for real roots), the possible integer values are $\lambda=1$ and $\lambda=2$.

Solving for $\lambda$ (Complex Roots)

Complex roots occur when the discriminant $D = 400 - 144\lambda < 0$, meaning $\lambda > \frac{25}{9} \approx 2.777...$. For complex conjugate roots, the magnitude of the difference $|\beta - \alpha|$ is given by $\frac{\sqrt{|D|}}{|a|} = \frac{\sqrt{144\lambda - 400}}{12}$. Applying the condition $\frac{1}{2} \leq |\beta - \alpha| \leq \frac{3}{2}$:

$ \frac{1}{2} \leq \frac{\sqrt{144\lambda - 400}}{12} \leq \frac{3}{2} $

Multiplying by 12 gives $6 \leq \sqrt{144\lambda - 400} \leq 18$. Squaring yields $36 \leq 144\lambda - 400 \leq 324$. Adding 400 results in $436 \leq 144\lambda \leq 724$. Dividing by 144:

$ \frac{436}{144} \leq \lambda \leq \frac{724}{144} $

Simplifying the fractions gives:

$ \frac{109}{36} \leq \lambda \leq \frac{181}{36} $

Approximately, $3.027... \leq \lambda \leq 5.027...$. The integers in this range satisfying $\lambda > 2.777...$ are $\lambda=4$ and $\lambda=5$.

Summing All Possible $\lambda$ Values

The set of all possible integer values for $\lambda$ is the union of values from both the real and complex roots cases: $\{1, 2\} \cup \{4, 5\} = \{1, 2, 4, 5\}$.

The sum of these values is $1 + 2 + 4 + 5 = 12$.