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    How would I answer this:

    The equation (z^2)+bz+11=0, where b is a real number, has distinct non real complex roots. Find the range of possible values of b.

    This is FP1, exam board: Edexcel.
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    (Original post by Doctor1234)
    How would I answer this:

    The equation (z^2)+bz+11=0, where b is a real number, has distinct non real complex roots. Find the range of possible values of b.

    This is FP1, exam board: Edexcel.
    If the quadratic in z doesn't have real roots, what do you know about the discriminant? (from C1)
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    (Original post by Doctor1234)
    How would I answer this:

    The equation (z^2)+bz+11=0, where b is a real number, has distinct non real complex roots. Find the range of possible values of b.

    This is FP1, exam board: Edexcel.
    Well, it's a quadratic with real coefficients. What's the condition on the discriminant that would imply complex roots for this equation?
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    Erm, we haven't covered discrimants yet.
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    (Original post by Doctor1234)
    Erm, we haven't covered discrimants yet.
    Should be like one of the first things in C1... maybe even at GCSE.

    The discriminant of a quadratic ax^2+bx+c=0 where a,b,c \in \mathbb \{ \mathbb{R}: a \neq 0 \} is given by b^2-4ac.

    Certain conditions on it determine the amount of real roots that the equation has.
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    (Original post by Doctor1234)
    Erm, we haven't covered discrimants yet.
    Okay, so I suppose you know the quadratic formula from GCSE:

    x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}.

    Now, the number inside the square root determines whether or not a quadratic has real solutions, since if the expression inside the square root is negative, then the square root will give a complex number.

    We call the bit inside the square root, ie b^2-4ac the discriminant.

    Can you work out what inequality the discriminant must satisfy for non-real roots, and hence answer the question?
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    (Original post by K-Man_PhysCheM)
    Okay, so I suppose you know the quadratic formula from GCSE:

    x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}.

    Now, the number inside the square root determines whether or not a quadratic has real solutions, since if the expression inside the square root is negative, then the square root will give a complex number.

    We call the bit inside the square root, ie b^2-4ac the discriminant.

    Can you work out what inequality the discriminant must satisfy for non-real roots, and hence answer the question?
    b^2 has to be less than 44 to obtain a non-real complex number?
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    (Original post by Doctor1234)
    b^2 has to be less than 44 to obtain a non-real complex number?
    Yes, so what range must b lie between?

    (Hint: sketch the parabola y=x^2 and look for the range of values where the curve lies below the line y=44. This will be a common technique for solving quadratic inequalities in A-level maths /FM)
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    (Original post by K-Man_PhysCheM)
    Yes, so what range must b lie between?

    (Hint: sketch the parabola y=x^2 and look for the range of values where the curve lies below the line y=44. This will be a common technique for solving quadratic inequalities in A-level maths /FM)
    As long as y is greater than 0 and less than 44?
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    (Original post by Doctor1234)
    As long as y is greater than 0 and less than 44?
    No, you need to sketch the parabola.

    Let's suppose you had b^2 < 4

    Then sketching y=x^2 and y=4 shows me that the curve is underneath the line between -2 and 2. So in this case, -2<b<2.

    Use the same for your question.
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    So in my case the curve would be underneath the line + or - root 44?
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    (Original post by Doctor1234)
    So in my case the curve would be underneath the line + or - root 44?
    Yeah, and that surd can be simplified and should be written as an inequality with b.
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    (Original post by RDKGames)
    Should be like one of the first things in C1... maybe even at GCSE.

    The discriminant of a quadratic ax^2+bx+c=0 where a,b,c \in \mathbb \{ \mathbb{R}: a \neq 0 \} is given by b^2-4ac.

    Certain conditions on it determine the amount of real roots that the equation has.
    Why do you use notation the OP probably won't understand?
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    (Original post by Ano9901whichone)
    Why do you use notation the OP probably won't understand?
    This notation is completely fine for FP1, there’s nothing OP should be unfamiliar with here
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    (Original post by K-Man_PhysCheM)
    Yeah, and that surd can be simplified and should be written as an inequality with b.
    I see, it makes sense. So it b is out of the range -(sqrt 44) and +(sqrt 44), then the equation would have real roots rather than imaginary roots?
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    (Original post by Protostar)
    This notation is completely fine for FP1, there’s nothing OP should be unfamiliar with here
    It looks a bit different than how I would format it on paper but yeah you need set notation for the things like solving sets of equations.
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    (Original post by Doctor1234)
    I see, it makes sense. So it b is out of the range -(sqrt 44) and +(sqrt 44), then the equation would have real roots rather than imaginary roots?
    Yes, that is right.

    Note it should be -2\sqrt{11} < b < 2\sqrt{11}

    and NOT  -2\sqrt{11} \leq b \leq 2\sqrt{11}

    because when b is exactly on the boundary, the quadratic would have one real repeated root.

    PS: it's good practice to always fully simplify surds
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    (Original post by K-Man_PhysCheM)
    Yes, that is right.

    Note it should be -2\sqrt{11} < b < 2\sqrt{11}

    and NOT  -2\sqrt{11} \leq b \leq 2\sqrt{11}

    because when b is exactly on the boundary, the quadratic would have one real repeated root.

    PS: it's good practice to always fully simplify surds
    My bad.
    Thanks for the help, cleared a lot of confusion. Are you doing A levels or are you at uni?
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    (Original post by Doctor1234)
    My bad.
    Thanks for the help, cleared a lot of confusion. Are you doing A levels or are you at uni?
    No worries

    I'm in Year 13, so not at Uni yet!
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    (Original post by K-Man_PhysCheM)
    No worries

    I'm in Year 13, so not at Uni yet!
    Oh kl, well good luck with A2.
 
 
 
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