Consider the polynomial \(\displaystyle{f{{\left({x}\right)}}}={x}^{{{2}}}-{5}{m}{x}+{10}{m}-{4}\), find m such

Pizzadililehz

Pizzadililehz

Answered question

2022-03-28

Consider the polynomial f(x)=x25mx+10m4, find m such that there exist a number a that satisfies f(a)=f(2a)=0.

Answer & Explanation

Ruben Gibson

Ruben Gibson

Beginner2022-03-29Added 9 answers

Let the roots be α,2α.
By Vieta's formulas,
Sum of roots =3α=5m, product of roots =2α2=10m4.
Hence, 2(5m3)2=10m4

25m245m+18=0
2(5m3)2=10m4

25m245m+18=0
we get (by simple factorisation) m=0.6 or 1.2.
Test by plugging these back into the original expression and getting f(x)=x23x+2 and f(x)=x26x+8 respectively, which each have two distinct roots with one twice the other.

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