Question

if alpha , beta are roots of x²+5x+a=0 and 2alpha + 5beta =-1, then find the value of a

Answer 1

Ans. is a = (-24) ... solution is in the pic :)

Answer 2

Given:

roots:

$\bold{x^2+5x+a=0}\\\bold{2\alpha +5 \beta =-1}$

To find:

a=?

Solution:

$x^2+5x+a=0.....(i)\\2\alpha +5 \beta =-1....(ii)$

Compare the value from the standard equation: $ax^2+bx+c=0$

$\to a= 1\\\to b=5\\\to c=a$

$\alpha+ \beta= \frac{-b}{a} \ \ \ \ and \ \ \ \ \alpha\beta= \frac{c}{a}\\\\(\alpha+ \beta)= -5 \ \ \ \ and \ \ \ \ \alpha \beta = a$

Solve the equation (ii):

$\bold{2\alpha +5 \beta =-1}\\\\\Rightarrow 2\alpha +3 \beta+2\beta =-1\\\\\Rightarrow 2\alpha +2\beta +3 \beta=-1\\\\\Rightarrow 2(\alpha +\beta) +3 \beta=-1\\\\\therefore \alpha +\beta= -5\\\\\Rightarrow 2(-5) +3 \beta=-1\\\\\Rightarrow -10 +3 \beta=-1\\\\\Rightarrow 3 \beta=-1+10\\\\\Rightarrow 3 \beta=9\\\\\Rightarrow \beta=\frac{9}{3}\\\\\Rightarrow \beta=3$

$\therefore \beta = 3 \ and \ \ \ (\alpha+ \beta)= -5\\\\\ So, \alpha = -8\\\\\Rightarrow \alpha+ \beta= -5\\\\\Rightarrow \alpha+ 3= -5\\\\\Rightarrow \alpha= -8$

roots $\alpha\ and \ \beta$ are -8 and 3.

$\therefore \alpha \beta= \frac{c}{a}\\\\\ where, c=a \ \ and a= 1\\\\ \alpha \beta= a\\\\\Rightarrow -8 \times 3= a\\\\\Rightarrow -24= a$

The final value of a is -24.