<p>Find the value of m for which the expression <strong>12x² − 10xy + 2y² − 11x − 5y + m</strong> can be resolved into two rational linear factors.</p>
Step-by-Step Solution
Key Concept: A quadratic expression in two variables resolves into linear factors if and only if the determinant condition abc + 2fgh − af² − bg² − ch² = 0 is satisfied.
<p><strong>Step 1:</strong> Identify coefficients by comparing with $ ax^2 + 2hxy + by^2 + 2gx + 2fy + c $:</p><p>a = 12, h = −5, b = 2, g = −$\frac{11}{2}$, f = −$\frac{5}{2}$, c = m</p><p><strong>Step 2:</strong> Apply the condition for resolution into linear factors:</p><p>$ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0 $</p><p><strong>Step 3:</strong> Substitute values:</p><p>$ 12 \cdot 2 \cdot m + 2\left(-\frac{5}{2}\right)\left(-\frac{11}{2}\right)(-5) - 12\left(-\frac{5}{2}\right)^2 - 2\left(-\frac{11}{2}\right)^2 - m(-5)^2 = 0 $</p><p><strong>Step 4:</strong> Simplify:</p><p>$ 24m - \frac{275}{2} - 75 - \frac{121}{2} - 25m = 0 $</p><p><strong>Step 5:</strong> Solve for m:</p><p>$ -m = -2 $ ∴ <strong>m = 2</strong></p>
Correct Answer: 2