Unknown Coefficents

Algebra Level pending

There are positive b b and c c such that the polynomial 2 x 2 + b x + c 2x^2+bx+c has two real roots which differ by 30. Find the least possible value of b + c b+c given b b and c c are positive integers.

This problem is not original.


The answer is 126.

Jason Hughes by Jason Hughes , 25, USA

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1 solution

Tom Engelsman
Feb 5, 2020

Rewrite the above quadratic equation as x 2 + b 2 x + c 2 = ( x + r ) ( x + ( r + 30 ) ) = 0. x^2 + \frac{b}{2}x + \frac{c}{2} = (x+r)(x + (r+30)) = 0. This yields: b = 4 r + 60 , c = 2 r 2 + 60 r b = 4r + 60, c = 2r^2 + 60r . Given that b , c ∈ N , b, c \in \mathbb{N}, we require:

b = 4 r + 60 > 0 ⇒ r > − 15 b = 4r+60 > 0 \Rightarrow r > -15 AND c = 2 r 2 + 60 r > 0 ⇒ r > 0 , r < − 30 c = 2r^2 + 60r > 0 \Rightarrow r > 0, r < -30

of which r > 0 r > 0 satisfies both conditions. Since r = 1 r = 1 is the smallest positive integer in this instance, this yields b = 64 , c = 62 ⇒ b + c = 126 . b = 64, c = 62 \Rightarrow b+c = \boxed{126}.

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