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Math Help - A few algebraic arithmetic questions that I'm stuck on....

  1. #1
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    A few algebraic arithmetic questions that I'm stuck on....

    Any help would be greatly appreciated I'm stuck on about 7 out of 30 questions - Here are a few:

    (i)
    Given that cx = \sqrt{\frac{ax^2 - b}{d}} express x in terms of a,b,c & d.

    Is this just a rearranging as I get a negative in my sqrt?

    (ii)
    If  x = p + \sqrt{q} where p and q are rational, show that x^2 and [tex]x^3/math] are of the form  P + Q\sqrt{q} where P and Q are rational.

    I'm stuck on this one completely

    (iii)
    Verify that  z = (4+\sqrt{15})^\frac{1}{3} + (4-\sqrt{15})^\frac{1}{3} satisfies  z^3 -3z +8 =0

    Not sure what is ment here...

    (iv)
    Given that \alpha and \beta are roots of the equation x^2 + 3x -6 =0 find a quadratic equation with integer coefficiants whose roots are \frac{2}{\alpha} and \frac{2}{\beta}

    I'm stuck with this and cannot find any examples of this online or in my text books.

    Positive is that I've gotten to grips with LaTex

    Thanks, Dojo
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  2. #2
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    For (i), the first thing you'll want to do is square both sides of the equation. Then you need to isolate x on one side of the equation. It pretty much is a rearranging.

    For (ii), just multiply out [Math](p + \sqrt{q})(p + \sqrt{q})[/tex] and note that a product or sum of rational numbers is rational.
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  3. #3
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    Quote Originally Posted by dojo View Post

    (iii)
    Verify that  z = (4+\sqrt{15})^\frac{1}{3} + (4-\sqrt{15})^\frac{1}{3} satisfies  z^3 -3z +8 =0

    Not sure what is ment here...
    This means if you expand

     \displaystyle \left( (4+\sqrt{15})^\frac{1}{3} + (4-\sqrt{15})^\frac{1}{3}\right)^3 -3\left( (4+\sqrt{15})^\frac{1}{3} + (4-\sqrt{15})^\frac{1}{3}\right) +8

    then simplify you should get zero.

    Or just solve z^3 -3z +8 =0

    What are the solutions??
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