Question

The condition for one zero of the quadratic polynomial a * x ^ 2 + bx + c to be twice the other is​

Answer 1

Let α and β are the two roots of the quadratic polynomial ax²+bx+c.

Then we know the conditions:

• $\sf{\alpha+\beta=\dfrac{-b}{a}}$

• $\sf{\alpha\beta=\dfrac{c}{a}}$

From the equation:

• $\sf{\beta=2\alpha}$

Case 1:

$\sf{\implies \alpha+2\alpha=\dfrac{-b}{a}}$

$\sf{\implies 3\alpha=\dfrac{-b}{a}}$

$\sf{\implies \alpha=\dfrac{-b}{3a} \ \ \ \ \ \longrightarrow(1)}$

Case 2:

$\sf{\implies \alpha\times 2\alpha=\dfrac{c}{a}}$

$\sf{\implies 2\alpha^2=\dfrac{c}{a}}$

$\sf{\implies \alpha^2=\dfrac{c}{2a} \ \ \ \ \ \longrightarrow(2)}$

Comparing (1) and (2):

$\sf{\implies (\dfrac{-b}{3a})^2=\dfrac{c}{2a}}$

$\sf{\implies \dfrac{b^2}{9a^2}=\dfrac{c}{2a}}$

$\sf{\implies \dfrac{b^2}{9a}=\dfrac{c}{2}}$

$\sf{\implies 2b^2=9ac}$

Therefore that is the condition for one zero of the quadratic polynomial ax²+bx+c to be twice the other.

$\huge{\boxed{\sf{ 2b^2=9ac}}}$