An easy one

Algebra Level 3

Find minimum value of a + 2 b + 3 c a+2b+3c , if one root of equation a x 2 + b x + c = 0 ax^2+bx+c=0 is also a root of equation x 2 + 2 x + 3 = 0 x^2+2x+3=0 .

Note: a , b a, b and c c are natural numbers.


The answer is 14.

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Ved Sharda by ved sharda , 20, India

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2 solutions

Ved Sharda
Jan 21, 2016

Equation x²+2x+3=0. has D<0 , so both imaginary roots are common as imaginary roots appear in conjugate pair

When both roots are common then : (a/1) = (b/2) = (c/3)

By this : b=2a and c=3a

So a+2b+3c = a+4a+9a = 14a

Given : a is natural number , so lowest natural number =1 , so a=1

So minimum value of a+2b+3c=14

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Irvine Dwicahya
Jan 28, 2016

Just think x^2 + 2x + 3 , its same thing with ax^2 + bx + c, And subtitution for a = 1, b = 2, c = 3. Soo a + 2b + 3c = (1)+2(2)+3(3) = 14

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