Question

If 3 : 1 is the ratio of the roots of the quadratic equation 2x2 + 5x + m = 0, then the value of m is equal to

Answer 1

Given:

Roots of Quadratic Equation $2x^2 + 5x + m = 0$ are 3 : 1.

To find:

m = ?

Solution:

Let the roots of the given quadratic equation be $\alpha$ and $\beta$.

As per given statement:

$\alpha : \beta = 3:1$

$\Rightarrow \dfrac{\alpha}{\beta } =\dfrac{3}{1}\\\Rightarrow \alpha=3\times \beta ...... (1)$

As per rule, the sum of roots of a quadratic equation $Ax^2 + Bx + C = 0$ is given by: $-\dfrac{B}{A}$

Here, A = 2, B = 5

As per above rule:

$\alpha +\beta =-\dfrac{5}{2} ...... (2)$

Putting value of $\alpha$ in equation (2) from equation (1):

$3 \times \beta +\beta =-\dfrac{5}{2} \\\Rightarrow \beta = -\dfrac{5}{8}$

By equation (1):

$\alpha = 3 \times \beta = -\dfrac{15}{8}$

Now, product of roots is given as: $\dfrac{C}{A}$

$\Rightarrow \alpha \beta = \dfrac{m}{2}\\\Rightarrow -\dfrac{5}{8}\times -\dfrac{15}{8} = -\dfrac{m}{2}\\\Rightarrow m =-\dfrac{75}{32}$

$\therefore \bold{m =-\dfrac{75}{32}}$