<p>Determine the values of <em>m</em> for which the equations <em>3x</em><sup>2</sup> + 4<em>mx</em> + 2 = 0 and <em>2x</em><sup>2</sup> + 3<em>x</em> − 2 = 0 may have a common root.</p>
Step-by-Step Solution
Key Concept: If two quadratic equations share a common root α, then α must satisfy both equations simultaneously. Use this condition to eliminate x and solve for m by setting up a system where the common root satisfies both equations.
<p><strong>Step 1:</strong> Let α be the common root. Then:</p><p>3α² + 4mα + 2 = 0 ... (1)</p><p>2α² + 3α − 2 = 0 ... (2)</p><p><strong>Step 2:</strong> From equation (2): 2α² = 2 − 3α, so α² = 1 − (3α/2)</p><p><strong>Step 3:</strong> Substitute into equation (1):</p><p>3(1 − 3α/2) + 4mα + 2 = 0</p><p>3 − (9α/2) + 4mα + 2 = 0</p><p>5 + α(4m − 9/2) = 0</p><p><strong>Step 4:</strong> This gives: α = −10/(8m − 9) ... (3)</p><p><strong>Step 5:</strong> From equation (2): 2α² + 3α − 2 = 0</p><p>Substitute α from (3) and solve for m:</p><p>2[−10/(8m − 9)]² + 3[−10/(8m − 9)] − 2 = 0</p><p>200/(8m − 9)² − 30/(8m − 9) − 2 = 0</p><p><strong>Step 6:</strong> Multiply by (8m − 9)²:</p><p>200 − 30(8m − 9) − 2(8m − 9)² = 0</p><p>200 − 240m + 270 − 2(64m² − 144m + 81) = 0</p><p>470 − 240m − 128m² + 288m − 162 = 0</p><p>−128m² + 48m + 308 = 0</p><p>128m² − 48m − 308 = 0</p><p>32m² − 12m − 77 = 0</p><p><strong>Step 7:</strong> Using quadratic formula: m = (12 ± √(144 + 9856))/64 = (12 ± √10000)/64 = (12 ± 100)/64</p><p>∴ m = 112/64 = 7/4 or m = −88/64 = −11/8</p>
Correct Answer: m = -11/8 or m = 7/4