Sum of Roots

Algebra Level 3

2 x 2 − 4 x + 1 = 0 \large 2x^{2} - 4x + 1 = 0

If α \alpha and β \beta are the roots of the equation above, then what is the value of 1 α + 2 β + 1 β + 2 α \frac{1}{ \alpha + 2\beta } + \frac{1}{ \beta + 2\alpha } ?

6 5 \frac{6}{5} 12 13 \frac{12}{13} 12 17 \frac{12}{17} 8 15 \frac{8}{15}

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Sakanksha Deo by Sakanksha Deo , 23, India

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1 solution

Using Vieta's formula we know that:

α + β = 2 α β = 1 2 \alpha+\beta=2 \\ \alpha \beta=\dfrac{1}{2}

Now, rewrite the expression:

1 α + 2 β + 1 β + 2 α = 3 ( α + β ) 5 α β + 2 ( α 2 + β 2 ) \dfrac{1}{\alpha+2\beta}+\dfrac{1}{\beta+2\alpha}=\dfrac{3(\alpha+\beta)}{5\alpha\beta+2(\alpha^2+\beta^2)}

And using ( α + β ) 2 = α 2 + 2 α β + β 2 (\alpha+\beta)^2=\alpha^2+2\alpha\beta+\beta^2 , then:

3 ( α + β ) α β + 2 ( α + β ) 2 \dfrac{3(\alpha+\beta)}{\alpha\beta+2(\alpha+\beta)^2}

Substituting the known values we get:

3 ( 2 ) 1 2 + 2 ( 2 ) 2 = 12 17 \dfrac{3(2)}{\dfrac{1}{2}+2(2)^2}=\boxed{\dfrac{12}{17}}

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Very good solution

Sai Ram - 5 years, 11 months ago

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