<p>For the quadratic equation 6<i>x</i><sup>2</sup> − 11<i>x</i> + <i>a</i> = 0, find the number of integer values of <i>a</i> for which the roots are rational.</p>
Step-by-Step Solution
Key Concept: The discriminant must be a perfect square for rational roots. Systematically check integer values of the parameter.
<p><strong>Step 1:</strong> For roots to be rational, the discriminant D = <i>b</i><sup>2</sup> − 4<i>ac</i> must be a perfect square.</p><p><strong>Step 2:</strong> For the equation 6<i>x</i><sup>2</sup> − 11<i>x</i> + <i>a</i> = 0:</p><p>$$D = (−11)^2 − 4(6)(a) = 121 − 24a$$</p><p><strong>Step 3:</strong> Check values of <i>a</i>:</p><p>D(1) = 121 − 24 = 97 (not a perfect square)</p><p>D(2) = 121 − 48 = 73 (not a perfect square)</p><p>D(3) = 121 − 72 = 49 = 7<sup>2</sup> ✓</p><p>D(4) = 121 − 96 = 25 = 5<sup>2</sup> ✓</p><p>D(5) = 121 − 120 = 1 = 1<sup>2</sup> ✓</p><p>D(6) = 121 − 144 = −23 < 0 (imaginary roots)</p><p>∴ For <i>a</i> ≥ 6, D < 0, hence imaginary roots.</p><p>Therefore, there are <strong>3 values</strong> of <i>a</i> (namely 3, 4, 5) for which roots are rational.</p>
Correct Answer: 3