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

1.	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» 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)) \(\Rightarrow\) \(\frac{3b^2}{16a^2}\) \(=\frac ca\) \(\Rightarrow\) \(3b^2=\frac{16a^2c}a=16ac\) Correct option is B) 3b²= 16ac

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»

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))

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

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

Correct option is B) 3b²= 16ac