Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app

Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

If x= 2 + 3 + 6 is a root of x^4+a x^3+b x^2+c x+d=0 where a, b, c,...

If

x=\sqrt{2}+\sqrt{3}+\sqrt{6}

is a root of

x^4+a x^3+b x^2+c x+d=0

where

a, b, c, d

are integers, what is the value of

|\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}|

?

Correct answer

93

Step-by-step solution

x=\sqrt{2}+\sqrt{3}+\sqrt{6}

root of equation :

x^4+a x^3+b x^2+c x+d=0 ; a, b, c, d \in Z

Now,

\begin{aligned} & (x-\sqrt{2})^2=(\sqrt{3}+\sqrt{6})^2 \Rightarrow \quad x^2-2 \sqrt{2} x+2=9+6 \sqrt{2} \\ \Rightarrow \quad & x^2-7=2 \sqrt{2}(3+x) \end{aligned}

(On squaring both sides)

\begin{array}{ll}\Rightarrow & \mathrm{x}^4-14 \mathrm{x}^2+49=8\left(\mathrm{x}^2+6 \mathrm{x}+9\right) \\ \Rightarrow & \mathrm{x}^4-22 \mathrm{x}^2-48 \mathrm{x}-23=0 \\ \therefore & a=0, b=-22, c=-48 \text { and } d=-23 \\ \therefore & |a+b+c+d|=93\end{array}

Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

Suppose the prime numbers p and q satisfy q^2+3 p=197 p^2+q . Write (q)/(p) as l+(m)/(n)… Olympiad workbook Let X=\ -5,-4,-3,-2,-1,0,1,2,3,4,5\ and S=\ (a, b) ∈ X × X: x^2+a x+b. and x^3+b x+a have… Olympiad workbook Suppose that P is the polynomial of least degree with integer coefficients such that P( 7… Olympiad workbook Let A=\ m: m. an integer and the roots of x^2+m x + 2020 = 0 are positive integers) and… Olympiad workbook Let x, y, z be complex numbers such that aligned & (x)/(y+z)+(y)/(z+x)+(z)/(x+y)=9 &… Olympiad workbook Suppose a, b are positive real numbers such that a a +b b =183, a b +b a =182 . Find… Olympiad workbook Let f(x)=x^2+a x+b . If for all nonzero real x f(x+(1)/(x))=f(x)+f((1)/(x)) and the roots… Olympiad workbook Let t be the area of a regular pentagon with each side equal to 1 . Let P(x)=0 be the… Olympiad workbook Full Quadratic Equation list All Olympiad workbook PYQs