Maths function interval

I have a question :

If x is real and x^2+(2-k)x+1-2k = 0

show that k cannot lie between certain limits, and find these limits.

After some working out I have a quadratic that factorises into two roots: k = -4 or k = 0.

I know this is asking for an interval but I don't know which interval it wants.

Is k < -4 and k > 0 ? Or is k > -4 and k < 0 ? It's the words 'can't lie between' which are puzzling me.

Thanks

Reply 1

mqb2766

Original post

by Skybird

I have a question :
If x is real and x^2+(2-k)x+1-2k = 0
show that k cannot lie between certain limits, and find these limits.
After some working out I have a quadratic that factorises into two roots: k = -4 or k = 0.
I know this is asking for an interval but I don't know which interval it wants.
Is k < -4 and k > 0 ? Or is k > -4 and k < 0 ? It's the words 'can't lie between' which are puzzling me.
Thanks

It helps to see your working but Im guessing you formed the discriminant and you should reason about which values of k give a value < 0.

You could sketch the parabola in terms of k (using the roots) and it should be fairly obvious which interval you want. Really youre just thinking about the sign of the k^2 coefficient.

Reply 2

ViviPanda

Original post

by Skybird

I have a question :
If x is real and x^2+(2-k)x+1-2k = 0
show that k cannot lie between certain limits, and find these limits.
After some working out I have a quadratic that factorises into two roots: k = -4 or k = 0.
I know this is asking for an interval but I don't know which interval it wants.
Is k < -4 and k > 0 ? Or is k > -4 and k < 0 ? It's the words 'can't lie between' which are puzzling me.
Thanks

First, to ensure that x is real, the quadratic equation must have real roots. This means the discriminant must be non-negative.
The given quadratic equation is: x^2 + (2-k)x + (1-2k) = 0
The discriminant Δ of a quadratic equation ax2+bx+c=0 is given by: Δ= b^2 - 4ac
For our equation:

•

a=1

•

b=2−k

•

c=1−2k

Substitute these values into the discriminant formula
For x to be real, the discriminant must be non-negative
Factor the quadratic
The product k(k+4)≥0 holds if k≤−4 or k≥0.
Therefore, k cannot lie between −4 and 0. The limits between which k cannot lie are −4 and 0.

Reply 3

Nitrotoluene

To ensure the quadratic equation [x^2] has real roots, you must calculate the values of (k) that are not allowed. The Delta of a quadratic equation must be ≥ 0 for its roots to be real.
Operatin on Delta with (a = 1), (b = 2 - k), and (c = 1 - 2k), you reach tis final result: (k^2 + 4k) ≥ 0). The latter is satisfied when (k ≤ -4) or (k > 0). So, (k) cannot stay within the range of (-4) and (0).
Bye,
Sandro
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