Question

If α, β are the zeroes of the polynomials f(x)= x²-3x+6, then find the value of 1/α + 1/β + α² +β² -2αβ.

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Answer 1

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Answer 2

Answer:

The value of $\frac{1}{\alpha} +\frac{1}{\beta} + \alpha^2 +\beta^2 -2\alpha\beta=-14.5$

Step-by-step explanation:

Given : If $\alpha,\beta$ are the zeroes of the polynomials $f(x)=x^{2} -3x+6$

To find : The value of $\frac{1}{\alpha} +\frac{1}{\beta} + \alpha^2 +\beta^2 -2\alpha\beta$

Solution :

The zeros of the quadratic equation $y=ax^2+bx+c$ is defined as:

Product of zeros $\alpha\beta=\frac{c}{a}$

Sum of zeros $\alpha+\beta=-\frac{b}{a}$

Comparing with given function the zeros are,

$\alpha\beta=\frac{6}{1}=6$  .....[1]

$\alpha+\beta=-\frac{-3}{1}=3$ ......[2]

Now, we solve the given value,

$=\frac{\beta+\alpha}{\alpha\beta}+\alpha^2 +\beta^2 -2\alpha\beta+2\alpha\beta-2\alpha\beta$

$=\frac{\beta+\alpha}{\alpha\beta}+(\alpha+\beta)^2-4\alpha\beta$

Substitute the value from [1] and [2],

$=\frac{3}{6}+(3)^2-4(6)$

$=\frac{1}{2}+9-24$

$=\frac{1}{2}-15$

$=-14.5$

Therefore, The value of $\frac{1}{\alpha} +\frac{1}{\beta} + \alpha^2 +\beta^2 -2\alpha\beta=-14.5$