Question

If one root of the equation ax² + bx + c = 0 is three times the other, then ……………

A) b² = 16ac 

B) 3b² = 16ac 

C) 2b² = 9ac 

D) b² = ac

Answer 1

Correct option is (B) 3b² = 16ac

Let \(\alpha\;\&\;3\alpha\) are roots of the quadratic equation \(ax^2+bx+c=0.\)

\(\therefore\) Sum of roots \(=\frac{-b}a\)

\(\Rightarrow\) \(\alpha+3\alpha\) \(=\frac{-b}a\)

\(\Rightarrow\) \(\alpha\) \(=\frac{-b}{4a}\)          _______________(1)

Product of roots \(=\frac ca\)

\(\Rightarrow\) \(\alpha.3\alpha\) \(=\frac ca\)

\(\Rightarrow\) \(3\alpha^2\) \(=\frac ca\)

\(\Rightarrow\) \(3(\frac{-b}{4a})^2\) \(=\frac ca\)            (From (1))

\(\Rightarrow\) \(\frac{3b^2}{16a^2}\) \(=\frac ca\)

\(\Rightarrow\) \(3b^2=\frac{16a^2c}a=16ac\)

Answer 2

Correct option is B) 3b² = 16ac