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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.
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)
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?
Reply 3
Erm, we haven't covered discrimants yet.
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 ax2+bx+c=0ax^2+bx+c=0 where a,b,c{R:a0}a,b,c \in \mathbb \{ \mathbb{R}: a \neq 0 \} is given by b24acb^2-4ac.

Certain conditions on it determine the amount of real roots that the equation has.
Original post by Doctor1234
Erm, we haven't covered discrimants yet.


Okay, so I suppose you know the quadratic formula from GCSE:

x=b±b24ac2ax = \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 b24acb^2-4ac the discriminant.

Can you work out what inequality the discriminant must satisfy for non-real roots, and hence answer the question?
Reply 6
Original post by K-Man_PhysCheM
Okay, so I suppose you know the quadratic formula from GCSE:

x=b±b24ac2ax = \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 b24acb^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?
Original post by Doctor1234
b^2 has to be less than 44 to obtain a non-real complex number?


Yes, so what range must bb lie between?

(Hint: sketch the parabola y=x2y=x^2 and look for the range of values where the curve lies below the line y=44y=44. This will be a common technique for solving quadratic inequalities in A-level maths /FM)
Reply 8
Original post by K-Man_PhysCheM
Yes, so what range must bb lie between?

(Hint: sketch the parabola y=x2y=x^2 and look for the range of values where the curve lies below the line y=44y=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?
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 b2<4b^2 < 4

Then sketching y=x2y=x^2 and y=4y=4 shows me that the curve is underneath the line between -2 and 2. So in this case, 2<b<2-2<b<2.

Use the same for your question.
Reply 10
So in my case the curve would be underneath the line + or - root 44?
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. :smile:
Original post by RDKGames
Should be like one of the first things in C1... maybe even at GCSE.

The discriminant of a quadratic ax2+bx+c=0ax^2+bx+c=0 where a,b,c{R:a0}a,b,c \in \mathbb \{ \mathbb{R}: a \neq 0 \} is given by b24acb^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?
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 :smile:
Reply 14
Original post by K-Man_PhysCheM
Yeah, and that surd can be simplified and should be written as an inequality with b. :smile:


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?
Original post by Protostar
This notation is completely fine for FP1, there’s nothing OP should be unfamiliar with here :smile:


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.
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 211<b<211-2\sqrt{11} < b < 2\sqrt{11}

and NOT 211b211 -2\sqrt{11} \leq b \leq 2\sqrt{11}

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

PS: it's good practice to always fully simplify surds :smile:
Reply 17
Original post by K-Man_PhysCheM
Yes, that is right.

Note it should be 211<b<211-2\sqrt{11} < b < 2\sqrt{11}

and NOT 211b211 -2\sqrt{11} \leq b \leq 2\sqrt{11}

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

PS: it's good practice to always fully simplify surds :smile:


My bad.
Thanks for the help, cleared a lot of confusion. Are you doing A levels or are you at uni?
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 :smile:

I'm in Year 13, so not at Uni yet!
Reply 19
Original post by K-Man_PhysCheM
No worries :smile:

I'm in Year 13, so not at Uni yet!


Oh kl, well good luck with A2.

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