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Math Help - Quadratic Equation: Condition for Roots Given, Coefficient Must be Figured Out

  1. #1
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    Quadratic Equation: Condition for Roots Given, Coefficient Must be Figured Out

    The question is:

    If one of the roots of x<sup>2</sup> + ax + 4 = 0 is twice the other root, then the value of 'a' is?

    (A) -3(2)<sup>0.5</sup>
    (B) 8(2)<sup>0.5</sup>
    (C) (2)<sup>0.5</sup>
    (D) -2(2)<sup>0.5</sup>

    I have no idea what to do...please help.
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  2. #2
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    If a quadratic equation x^2+ax+b=0 has roots x_1,x_2, then x^2+ax+b=(x-x_1)(x-x_2). Also, two polynomials are equal iff their respective coefficients are equal.

    According to my calculations, the answer is "none of the above" because one variant is only a partial, not complete, answer.

    In typing polynomials in ASCII, it is customary to indicate power using ^, e.g.: x^2 + ax + 4. You can also use LaTeX: type [tex]x^2+ax+4[/tex] and [tex]\sqrt{2}[/tex] to get x^2+ax+4 and \sqrt{2}.
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  3. #3
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    Quote Originally Posted by quadrat View Post
    The question is:

    If one of the roots of x<sup>2</sup> + ax + 4 = 0 is twice the other root, then the value of 'a' is?

    (A) -3(2)<sup>0.5</sup>
    (B) 8(2)<sup>0.5</sup>
    (C) (2)<sup>0.5</sup>
    (D) -2(2)<sup>0.5</sup>

    I have no idea what to do...please help.
    1. Solve the equation

    x^2+ax+4=0 for x.

    You should come out with:

    x_1 = -\frac a2-\sqrt{\frac{a^2}4-4}~\vee~x_1 = -\frac a2+\sqrt{\frac{a^2}4-4}

    2. According to the text of the peoblem you know that

    2x_1 = x_2 . That means:

    2\cdot \left(-\frac a2-\sqrt{\frac{a^2}4-4}\right)= -\frac a2+\sqrt{\frac{a^2}4-4}

    Solve for a.

    3. Btw I've got 2 different values for a. But please check my calculations!
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  4. #4
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    Yet another variation: Let x_1 be one of the roots- the other is 2x_1. Then we have (x- x_1)(x- 2x_1)= x^2- (x_1+ 2x_1)+ 2x_1^2= x^2- 3x_1x+ 2x_1^2= x^2+ ax+ 4
    Now you know that 2x_1^2= 4 and a= -3x_1. Yes, there are two possible values for a but only one of them is one of the choices.
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  5. #5
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    Quote Originally Posted by quadrat View Post
    The question is:

    If one of the roots of x<sup>2</sup> + ax + 4 = 0 is twice the other root, then the value of 'a' is?

    (A) -3(2)<sup>0.5</sup>
    (B) 8(2)<sup>0.5</sup>
    (C) (2)<sup>0.5</sup>
    (D) -2(2)<sup>0.5</sup>

    I have no idea what to do...please help.
    If you can't write x^2, then please at least use x^2 so that what you post is more easily read etc.
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