Inequalities

[12345678]

‘Find the values of k for which the follow equation has twodistinct roots’
Kx² - 2Kx + 5 =0
If an equation has two distinct roots, b² - 4ac > 0
a = K, b = (-2K), c = 5
(-2K)² - (4 * K * 5) > 0
4K² - 20K > 0
4K (K – 5 ) > 0
K > 0
The other answer is k > 8.
However, I believe my working to get K > 0 (which is thecorrect answer) is wrong, as subbing k = 1 would give 4 * (-4) = -16 which isnot greater than 0.
Anybody spot where I’m going wrong?

 

[stapel]

‘Find the values of k for which the follow equation has twodistinct roots’
Kx² - 2Kx + 5 =0
If an equation has two distinct roots, b² - 4ac > 0
a = K, b = (-2K), c = 5
(-2K)² - (4 * K * 5) > 0
4K² - 20K > 0
4K (K – 5 ) > 0

This is a quadratic inequality. You need to find the intervals on which the parabola y = 4K(K - 5) is above the x- (or, properly, the K-) axis.

K > 0

No. If K = 1, then 4K(K - 5) = 4(1)((1) - 5) = 4(1)(-4) = -16 < 0, not greater than zero.

Inequalities do not solve in the same manner as do equations. You can learn the general methodology here. Find the x- (or K-) intercepts, and then figure out where the parabola is above the axis.

The other answer is k > 8.

How did you arrive at this interval?

Note: If K = 6, then 4K(K - 5) = 4(6)(6 - 5) = 24(1) = 24 > 0. So "K > 8" cannot be the proper bound.

 

[12345678]

This is a quadratic inequality. You need to find the intervals on which the parabola y = 4K(K - 5) is above the x- (or, properly, the K-) axis.


No. If K = 1, then 4K(K - 5) = 4(1)((1) - 5) = 4(1)(-4) = -16 < 0, not greater than zero.

Inequalities do not solve in the same manner as do equations. You can learn the general methodology here. Find the x- (or K-) intercepts, and then figure out where the parabola is above the axis.


How did you arrive at this interval?

Note: If K = 6, then 4K(K - 5) = 4(6)(6 - 5) = 24(1) = 24 > 0. So "K > 8" cannot be the proper bound.

Oh, so do I solve this by drawing the curve?
and the text book gave the answer k<0 (my mistake) and k >8

 

[stapel]

Oh, so do I solve this by drawing the curve?

Solving the inequality can be simplified by looking at the graph (or just picturing it in your head, because you know what a positive quadratic's graph looks like). The lesson at the link illustrates how this works.

and the text book gave the answer k<0 (my mistake) and k >8

As pointed out earlier, "K > 8" is not correct.

 

[Deleted member 4993]

Oh, so do I solve this by drawing the curve?
and the text book gave the answer k<0 (my mistake) and k >8

Your equation must have been:

Kx² - 2Kx + 8 =0