The quadratic equation whose roots are a/b and b/a, a≠b≠0,where a2=5a−3, and b2=5b−3 is

a, b are roots of x2-5x+3=0 x=5±132 ab=5+135-13; ba=5-135+13 ab+ba=5+135-13+5-135+13=2(25+13)25-13=7612=193 3x2−19x+3=0

The quadratic equation whose roots are a/b and b/a, a≠b≠0,where a2=5a−3, and b2=5b−3 is

a, b are roots of x2-5x+3=0 x=5±132 ab=5+135-13; ba=5-135+13 ab+ba=5+135-13+5-135+13=2(25+13)25-13=7612=193 3x2−19x+3=0

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The quadratic equation whose roots are a/b and b/a,
$a \neq b \neq 0,$ where $a^{2} = 5 a - 3,$ and $b^{2} = 5 b - 3$ is

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$3 x^{2} + 19 x - 3 = 0$

$3 x^{2} - 19 x + 3 = 0$

$3 x^{2} + 19 x + 3 = 0$

$3 x^{2} - 19 x - 3 = 0$

answer is A.

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Detailed Solution

$a, b a r e r o o t s o f x^{2} - 5 x + 3 = 0 x = \frac{5 \pm \sqrt{13}}{2} \frac{a}{b} = \frac{5 + \sqrt{13}}{5 - \sqrt{13}}; \frac{b}{a} = \frac{5 - \sqrt{13}}{5 + \sqrt{13}} \frac{a}{b} + \frac{b}{a} = \frac{5 + \sqrt{13}}{5 - \sqrt{13}} + \frac{5 - \sqrt{13}}{5 + \sqrt{13}} = \frac{2 (25 + 13)}{25 - 13} = \frac{76}{12} = \frac{19}{3} 3 x^{2} - 19 x + 3 = 0$