If α and β are zeroes of the quadratic polynomial x2 − 6x + a; find the value of 'a' if 3α + 2β = 20.
Given that: α and β are the zeroes of the quadratic polynomial x2 − 6x + a and 3α + 2β = 20, then we have to find the value of a.
We have the following procedure.
$\alpha+\beta=-\frac{-6}{1}$
$=6$...............(1)
$\alpha \beta=\frac{a}{1}$
$=a$.........(2)
$3 \alpha+2 \beta=20$
$\Rightarrow 2(\alpha+\beta)+\alpha=20$
$\Rightarrow \quad 2 \times 6+\alpha=20$
$\Rightarrow \quad \alpha=8$
Now, we are putting the value α in equation (1), we get
$8+\beta=6$
$\Rightarrow \quad \beta=-2$
From equation (2), we have
$a=8 \times(-2)$
$=-16$
Hence the value of a is −16.
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