Question

The quadratic equation those real coefficients whose one root is 5 + root 5 by 2 is

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

In an equation with real coefficients irrational roots occur in conjugate pairs

More precisely, if a +√b is a root of a quadratic equation then a - √b is another root of the equation

GIVEN

$\sf{One \: root \: of \: a \: quadratic \: equation \: with \: real}$

$\displaystyle \: \sf{ coefficients\: \: is \: \: \: \: \: 5 + \frac{ \sqrt{5} }{2} }$

TO DETERMINE

The quadratic equation

CALCULATION

$\sf{ Since\: One \: root \: of \: a \: quadratic \: equation \: with \: real}$

$\displaystyle \: \sf{ coefficients\: \: is \: \: \: \: \: 5 + \frac{ \sqrt{5} }{2} }$

$\displaystyle \: \sf{ \: \: 5 - \frac{ \sqrt{5} }{2} }$

$\displaystyle \: \sf{ \therefore \: Sum \: of \: the \: roots\: =5 + \frac{ \sqrt{5} }{2} + \: 5 - \frac{ \sqrt{5} }{2} = 10 }$

$\displaystyle \: \sf{ \therefore \: Product \: of \: the \: roots\: }$

$\displaystyle \: \sf{ = \bigg(5 + \frac{ \sqrt{5} }{2} \bigg) \bigg(5 - \frac{ \sqrt{5} }{2} \bigg) }$

$\displaystyle \: \sf{ = {(5)}^{2} - { \bigg(\frac{ \sqrt{5} }{2} \bigg)}^{2} }$

$\displaystyle \: \sf{ = 25 - \frac{5}{4} }$

$\displaystyle \: \sf{ = \frac{95}{4} }$

So the required Quadratic Equation is

$\sf{ {x}^{2} - (sum \: of \: the \: zeroes)x + (product \: of \: the \: zeroes ) = 0}$

$\displaystyle \: \sf{ {x}^{2} - 10x + \frac{95}{4} = 0 }$