Question

If 1 is a root of the quadratic equation 3x² + ax – 2 = 0 and the quadratic equation a(x² + 6x) – b = 0 has equal roots, find the value of b ?

Answer 1

Since 1 is a root of the quadratic equation 3x² + ax − 2 = 0, we have:
⇒ 3(1)² + a(1) − 2 = 0
⇒ 3 + a − 2 = 0
⇒ a = −1              ...(i)

It is given that the equation a(x² + 6x) − b = 0 has equal roots.
For roots to be equal, the discriminant of the quadratic equation should be 0.

\[\text{Now}, \left( 6a \right)^2 - 4(a)( - b) = 0\]

\[ \Rightarrow 36 a^2 + 4ab = 0\]

\[ \Rightarrow 4a\left( 9a + b \right) = 0\]

\[ \Rightarrow 9a + b = 0 \left( \because a \neq 0 \right)\]

\[ \Rightarrow b = - 9a\]

\[\text{On putting the value of a, we get}: \]

\[b = - 9\left( - 1 \right) = 9\]

Thus, the value of b is 9.