Find the possible values of k for the quadratic equation 2kx^2 + 5kx + 5k - 3 =0

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Aleksander Krol
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Find the possible values of k for the quadratic equation 2kx^2 + 5kx + 5k - 3 =0 to have real roots.
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so isn't the discriminant supposed to be b^2 -4ac >=0 over here?
but in the ms they just used greater than (>0) sign why?
and why is the curve a sad face? isn't it supposed to be a happy face because the coefficient of x^2 is positive?
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Apad121
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Hi there,

It's because if it is =0 it would have one "real root" while your question says "real roots" that's plural which means it must be >0

Hope this helps!
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Aleksander Krol
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(Original post by Apad121)
Hi there,

It's because if it is =0 it would have one "real root" while your question says "real roots" that's plural which means it must be >0

Hope this helps!
oh thanks i inderstood the first part now. but then i don't get why is the curve a sad face though?
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Apad121
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(Original post by Aleksander Krol)
oh thanks i inderstood the first part now. but then i don't get why is the curve a sad face though?
They are plotting to find the value of K. This is no longer the original equation they're plotting but the one beginning -15k² so it is a "sad" face. This is a standard method of finding quadratic inequalities.
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_gcx
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That solution is written very poorly and yes I'd read the question to mean at least one real root. They shouldn't turn the inequality to an equality like that. Instead you read the solution set to the inequality from the sketch (looking at where the curve lies above or on the x-axis) with the intersections 0 and 5/8 in mind.

Bear in mind the sketch is of 3k(-5k + 8) and not the original quadratic. The leading term is -15k^2 hence the "sad face" shape.
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Aleksander Krol
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(Original post by _gcx)
That solution is written very poorly and yes I'd read the question to mean at least one real root. They shouldn't turn the inequality to an equality like that. Instead you read the solution set from the sketch (looking at where the curve lies above or on the x-axis) with the intersections 0 and 5/8 in mind.

Bear in mind the sketch is of 3k(-5k + 8) and not the original quadratic. The leading term is -15k^2 hence the "sad face" shape.
so according to the question the discriminant is >=0 right?
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Aleksander Krol
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(Original post by Apad121)
They

They are plotting to find the value of K. This is no longer the original equation they're plotting but the one beginning -15k² so it is a "sad" face. This is a standard method of finding quadratic inequalities.
(Original post by Apad121)
Hi there,

It's because if it is =0 it would have one "real root" while your question says "real roots" that's plural which means it must be >0

Hope this helps!
wait,.... over here you said, "It's because if it is =0 it would have one "real root" while your question says "real roots" that's plural which means it must be >0" but when the discriminamt =0 it has 1 repeated real roots so it's still plural though.
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_gcx
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(Original post by Aleksander Krol)
so according to the question the discriminant is >=0 right?
I would say so, it's not very clear. An actual exam question would be clearer.
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Aleksander Krol
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(Original post by _gcx)
I would say so, it's not very clear. An actual exam question would be clearer.
thank you!! i was really scared cause i recently had a test on this chapter and rn after looking at this question's ms i felt like i really messed up my grades this time. 😂 so it's a relief to hear that. thanks again. :3 <3
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Apad121
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(Original post by Aleksander Krol)
wait,.... over here you said, "It's because if it is =0 it would have one "real root" while your question says "real roots" that's plural which means it must be >0" but when the discriminamt =0 it has 2 repeated real roots so it's still plural though.
It is standard in A level questions to be worded like this which can be frustrating.

One root is another way of saying repeating root because it only touches the X axis at one point. It isn't called "2 repeated roots" _formally but just "1 repeated root" so it shouldn't be plural.
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_gcx
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(Original post by Apad121)
It is standard in A level questions to be worded like this which can be frustrating.

One root is another way of saying repeating root because it only touches the X axis at one point. It isn't called "2 repeated roots" _formally but just "1 repeated root" so it shouldn't be plural.
I'd think they'd say "distinct roots" where that is expected. I don't remember it ever being ambiguous. (indeed most weird/badly asked questions in textbooks wouldn't make it to the final exam since they're checked much more carefully) I'd expect them to say "at least one real root" if they allowed a repeated root.

If it was ambiguous, they would/should accept both interpretations.
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mqb2766
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Agree with the above, but its always worth checking before you (the book) starts talking about a quadratic discriminant, that
a = 2k
(a is the x^2 coefficient) is not zero. Otherwise the quadratic formula is irrelevant (no quadratic term or divide by zero) and hence the discriminant is irrelevant as well. If you simply assume the discriminant / quadratic formula is valid, you might get into problems. In this case, if k=0 you have
-3 = 0
Its not even valid, never mind having any type of roots.
Last edited by mqb2766; 1 month ago
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