Sum, Product of Roots alpha, beta of 4x^2 - x + 7 = 0

DesperateLearner

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Really stuck forgot how to find alpha and beta and what to do next, please help me :( ..thanks! Hope the attachment works.

If \(\displaystyle \alpha\) and \(\displaystyle \beta\) are the roots of the quadratic equation \(\displaystyle 4x^2\, -\, x\, +\, 7\, =\, 0,\) find the value of each of the following:

. . .\(\displaystyle \mbox{i. }\, 4\alpha\, +\, 4\beta\)

. . .\(\displaystyle \mbox{ii. }\,\dfrac{\alpha}{\beta}\, +\, \dfrac{\beta}{\alpha}\)

. . .\(\displaystyle \mbox{iii. }\, \alpha^3\, +\, \beta^3\)

. . .\(\displaystyle \mbox{iv. }\, \alpha^4\, +\, \beta^4\)
 

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DesperateLearner;42986 2 said:
Really stuck forgot how to find alpha and beta and what to do next, please help me :( ..thanks! Hope the attachment works.

What are the roots of:

Ax^2 + Bx + C = 0
 
I would proceed as Subhotosh Khan suggested, and consider the general quadratic:

\(\displaystyle ax^2+bx+c=0\tag{1}\)

And develop theorems pertaining to this general quadratic to answer the questions in terms of the parameters \(\displaystyle a,\,b,\,c\) from which you can then answer for the specific quadratic given. So, what are the roots of (1)?
 
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