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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Find the number of ordered triples (x, y, z) of real numbers that...

Find the number of ordered triples

(x, y, z)

of real numbers that satisfy the system of equation :

x+y+z=7 ; x^2+y^2+z^2=27 ; x y z=5

Correct answer

03

Step-by-step solution

\begin{aligned} & x+y+z=7, x^2+y^2+z^2=27, x y z=5 \\ & x y+y z+z x=\frac{49-27}{2}=11 \end{aligned}

\therefore \quad

Cubic whose roots are

x, y, z

is

t^3-7 t^2+11 t-5=0

Its roots are 1,1,5

\Rightarrow \quad(x, y, z) \equiv(1,1,5)

So number of triplets

=\frac{3!}{2!}=3

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