Question

Alpha and beta are the zeros of the polynomial 2x2-4x+1 then find value of 1/alpha+2beta + 1/beta+2alpha​

Answer 1

Answer: 12/17

Step-by-step explanation:

Please see the attached explanation

Answer 2

The value of the given expression is 12/17

Step-by-step explanation:

The given quadratic equation is

$2x^2-4x+1$

For this equation, we have

a = 2, b = -4, c = 1

Now, it has been given that α and β are the zeros of the equation.

Hence, Sum of zeros is given by

$\alpha+\beta=-\frac{b}{a}\\\\\alpha+\beta=-\frac{-4}{2}\\\\\alpha+\beta=2...(i)$

And Product of zeros is given by

$\alpha\cdot\beta=\frac{c}{a}\\\\\alpha\cdot\beta=\frac{1}{2}...(ii)$

Now, we have to find the value of the expression

$\frac{1}{\alpha+2\beta}+\frac{1}{2\alpha+\beta}$

Simplify this rational expression in terms of α + β and αβ

$\frac{2\alpha+\beta+\alpha+2\beta}{\alpha\beta+2\alpha^2+2\beta^2+4\alpha\beta}\\\\=\frac{3(\alpha+\beta)}{5\alpha\beta+2(\alpha^2+\beta^2)}\\\\=\frac{3(\alpha+\beta)}{5\alpha\beta+2((\alpha+\beta)^2-2\alpha\beta)}$

Substituting the values α + β and αβ from (i) and (ii)

$\frac{3\cdot2}{5\cdot\frac{1}{2}+2((2)^2-2\cdot\frac{1}{2}}\\\\=\frac{6}{17/2}\\\\=\frac{12}{17}$

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