Question

The coefficient of quadratic equations ax^2+(a+d)X+(a+2d)=0 are consecutive terms of a positively valued, increasing AP. Determine the least integral value of d/a such that the equation has real roots.​

Answer 1

Answer is 7

$\sf </p><p>For\:equation\: to \:have\: real \:roots, </p><p>D>0 \\\\\sf</p><p>=b^2-4ac>0 \\\\\sf</p><p>=(a+d)^2-4a(a+2d) \\\\\sf</p><p>=a^2+d^2+2ad-4a^2-8ad \\\\\sf</p><p>=d^2-3a^2-6ad \\\\\sf</p><p>Dividing,\:=d^2-3a^2-6ad \:by \:a^2</p><p>We\:get, \\\\\sf</p><p>=(\frac{d}{a})^2-3-\frac{6d}{a} \\\\\sf</p><p>Let \frac{d} {a} = m \\\\\sf</p><p>=(\frac{d}{a})^2-3-\frac{6d}{a}=m^2-6m-3 \\\\\sf</p><p>By,\:Brahmagupta's \:Quadratic\: formula \\\\\sf</p><p>=\frac{-b±\sqrt{b^2-4ac}}{2a} \\\\\sf</p><p>=\frac{6±\sqrt{36-12}}{2} \\\\\sf </p><p>=\frac{6±\sqrt{24}}{2} \\\\\sf </p><p>=3±\sqrt{12} \\\\\sf </p><p>=3±2\sqrt{3} \\\\\sf </p><p>m\:\epsilon(-∞, 3-2\sqrt{3}) U(3+2\sqrt{3}, ∞) \\\\\sf</p><p>\frac{d} {a} \:\epsilon(-∞, 3-2\sqrt{3}) U(3+2\sqrt{3}, ∞) \\\\\sf</p><p>The\: least\: integral\: value \:of \:\frac{d} {a} \\\\\sf</p><p>=3+2\sqrt{3} \\\\\sf</p><p>=3+2x1.73 \\\\\sf</p><p>=3+3.46</p><p>=6.46 \\\\\sf</p><p>Integral\: value=7</p><p>$