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Thread: Roots ><

  1. #1
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    Roots ><

    hey, ive got a few Qs here. hope someone could help me with them =)

    1) The roots of the quadratic equation are $\displaystyle 2x^2 - 3x +7 = 0 $ are $\displaystyle \alpha and \beta$
    i) Find the quadratic equation in X whose roots are $\displaystyle \frac{\alpha}{\beta} and \frac{\beta}{\alpha}$

    I dont need the answer to this, i just wanna know what they mean by quadratic equation in X?

    2) The roots of the quadratic equation $\displaystyle 2x^2 - 6c + 5 = 0$ are $\displaystyle \alpha and \beta$.
    Find the quadratic equation with interger coefficients whose roots are $\displaystyle \frac{2}{\alpha} and \frac{2}{\beta}$

    I dont need the answer to this, i just wanna know what they mean by quadratic equation with integer coefficients?

    3) Given that $\displaystyle \alpha and \beta$ are the roots of the equation $\displaystyle x^2 - 3x = 5$, find the value of $\displaystyle \frac{1}{\alpha^2} + \frac{1}{\beta^2} $ and show that $\displaystyle \alpha^4 = 57\alpha + 70$

    4) The roots of the equation $\displaystyle x^2 - \sqrt{13x} + 1 = 0$ are p and q, where p>q. Without using a calculator, express $\displaystyle \frac{p}{q}$ in the form of $\displaystyle \frac{a - \sqrt{b}}{c}$, where a,b and c are prime numbers.

    5) The roots of the equation $\displaystyle x^2 + mx + n = 0$ are $\displaystyle \alpha$ and $\displaystyle \beta$, where m and n are real constants. Express the following expression in terms of m and n.
    if $\displaystyle \alpha = \beta - 1$, show that $\displaystyle 4n = m^2 -1$

    Really hope someone could help me. my exam is tomorrow and i really wanna know the answers by today.
    Thanks in advance!!^^
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  2. #2
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    I am no fully clear as to what is required here. But this may help you.
    The numbers a & b are roots of $\displaystyle (x-a)(x-b)=x^2 - (a+b) + ab = 0$.
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  3. #3
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    hey plato!
    thanks for replying! ^^
    and erm,i dont really get what you mean by
    $\displaystyle
    (x-a)(x-b)=x^2 - (a+b) + ab = 0
    $

    sorry, my maths is really bad ><
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by teddybear67 View Post
    hey plato!
    thanks for replying! ^^
    and erm,i dont really get what you mean by
    $\displaystyle
    (x-a)(x-b)=x^2 - (a+b) + ab = 0
    $

    sorry, my maths is really bad ><
    What Plato is saying is: If a and b are roots (zeros) of the quadratic equation, then (x - a) and (x - b) are factors. Therefore,

    (x -a )(x - b) = 0
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