if one root of the euation is ax^+bx^+c= 0is triple the other show that 3b²=16bc.
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ax²+bx+c=0 --------------(1)
Let α,β are the roots of (1). Then by the given question,
α=3β
From the relation between roots and coefficients,
α+β=-b/a
or, 3β+β=-b/a
or, 4β=-b/a
or, β=-b/4a ---------------(2) and
α×β=c/a
or, 3β×β=c/a
or, 3β²=c/a
or, 3×(-b/4a)²=c/a
or, 3b²/16a²=c/a
or, 3b²=16a²×c/a
or, 3b²=16ac
Let α,β are the roots of (1). Then by the given question,
α=3β
From the relation between roots and coefficients,
α+β=-b/a
or, 3β+β=-b/a
or, 4β=-b/a
or, β=-b/4a ---------------(2) and
α×β=c/a
or, 3β×β=c/a
or, 3β²=c/a
or, 3×(-b/4a)²=c/a
or, 3b²/16a²=c/a
or, 3b²=16a²×c/a
or, 3b²=16ac
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