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Math Help - some simple questions.from my exam(roots ,partial fractions ..)

  1. #1
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    some simple questions.from my exam(roots ,partial fractions ..)

    hi guys i had an exam today and just wanted you guys to give me the correct answer for it so i can see how i went.
    i'm sure its really easy for you guys

    1) if \alpha and  \beta are the roots of the equation x^2 - 5x + 4 = 0 find
    i) \alpha^2 + \beta^2 ii) \alpha^2 \beta^2 iii) the equation with roots \alpha^2 and \beta^2


    2) the curve  f(x) = ax^2 + bx + c has factors  (x + 1) and  (x + 3) and cuts the y-axis at y = 8 . Find the values of a,b,c

    3) Express in partial fractions  \frac{2}{(x-2)(2x^2+1)} in partial fractions.

    4) Simplify  \frac{1}{\sqrt{x+1}}+\sqrt{x}, rationalising the denominator.

    thanks for the help!
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  2. #2
    Super Member dhiab's Avatar
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    Hello : Question 1


    the equation is :
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  3. #3
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    Hello, llkkjj24!


    1) if \alpha and \beta are the roots of the equation: x^2 - 5x + 4 \:=\: 0, find:

    (i)\;\;\alpha^2 + \beta^2
    We know that: . \begin{array}{cc}\alpha + \beta \:=\:5 & [1] \\ \alpha\!\cdot\!\beta \:=\:4 & [2] \end{array}


    \text{Square [1]: }\;(\alpha + \beta)^2 \:=\:5^2 \quad\Rightarrow\quad \alpha^2 + 2(\alpha\beta) + \beta^2 \:=\:25
    . . . . . . . . . . . . . . . . . . . . . . . . . . \uparrow
    . . . . . . . . . . . . . . . . . . . . . . . . ^{\text{This is 4}}

    And we have: . \alpha^2 + 8 + \beta^2 \:=\:25 \quad\Rightarrow\quad \alpha^2 + \beta^2 \:=\:17




    (ii)\;\;\alpha^2 \beta^2
    Since \alpha\beta \:=\:4, then: . \alpha^2\beta^2 \:=\:16



    (iii) the equation with roots \alpha^2 and \beta^2
    The equation is: . (x - \alpha^2)(x - \beta^2) \:=\:0 \quad\Rightarrow\quad x^2 - \alpha^2x - \beta^2x + \alpha^2\beta^2\:=\:0

    . . x^2 - \underbrace{(\alpha^2 + \beta^2)}x + \underbrace{\alpha^2\beta^2} \:=\:0
    . . . . . . . . \uparrow . . . . . . \uparrow
    . . . . . . ^{\text{This is 17}} . . ^{\text{This is 16}}


    Therefore: . x^2 - 17x + 16 \;=\;0

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  4. #4
    Super Member dhiab's Avatar
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    Quote Originally Posted by llkkjj24 View Post
    hi guys i had an exam today and just wanted you guys to give me the correct answer for it so i can see how i went.
    i'm sure its really easy for you guys

    1) if \alpha and  \beta are the roots of the equation x^2 - 5x + 4 = 0 find
    i) \alpha^2 + \beta^2 ii) \alpha^2 \beta^2 iii) the equation with roots \alpha^2 and \beta^2


    2) the curve  f(x) = ax^2 + bx + c has factors  (x + 1) and  (x + 3) and cuts the y-axis at y = 8 . Find the values of a,b,c

    3) Express in partial fractions  \frac{2}{(x-2)(2x^2+1)} in partial fractions.

    4) Simplify  \frac{1}{\sqrt{x+1}}+\sqrt{x}, rationalising the denominator.

    thanks for the help!
    HELLO : QUESTION 2
    Ihave the system
    Continu......
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