[ASK]Quadratic Inequation

Monox D. I-Fly · 2019-10-03 14:27:17

How to solve this?

This is what I've done so far:
9285d1570155869-ask-quadratic-inequation-takokke-forum-jpg

How to solve the quadratic inequation formed? Factorization is an obvious no, and the quadratic formula will just make everything messier, is there any other way?

Bob · 2019-10-03 19:24:10

hi Monox D. I-Fly

There's a formula that will convert a log in any base into a log in a different base. You'll have to look it up as I've forgotten the exact form.

Then you can change all the logs into log base 2.

log(x) can then be cancelled out leaving a simple equation. But beware, you need to consider three cases separately. What if log(x) is less than 0, equal to zero, and greater than zero.

I'm off out now but I'll give it a try myself later on.

Bob

Monox D. I-Fly · 2019-10-03 19:57:45

I'll look forward to what you'll come up with.

Also, thanks for removing the useless comment from another member.

Bob · 2019-10-04 01:56:33

Edited for final solution.

Firstly, the x^2 term can be changed to this:

log x^2 = 2.log x

The formula to change a log base is worked out as follows:

Take logs in base b

Therefore

Change all the logs to base 10 (chosen so I can use a calculator later).

I'll call the number in the bracket E to save typing.

Can you complete it from here?

I cannot; I'm getting no real roots ???

Bob

Monox D. I-Fly · 2019-10-04 14:24:14

So, there's indeed no real solution?

Bob · 2019-10-05 05:18:14

Unless I've slipped up somewhere. Anyone else care to check this?

Bob

Bob · 2019-10-05 20:16:15

I've plotter the graph and it's a U-shaped quadratic above the x axis. So no roots seems right. But as we want >0 we can say the inequality is true for all Y

That is

Bob

Monox D. I-Fly · 2019-10-06 14:02:34

bob bundy wrote:

I've plotter the graph and it's a U-shaped quadratic above the x axis. So no roots seems right. But as we want >0 we can say the inequality is true for all Y
That is
Bob

Well, my partner (who asked the question in the first place) did try an online calculator to solve it and got no result, but she thought that it's merely because no one has submitted a solution to the exact question yet.

Bob · 2019-10-07 01:49:19

The problem I've had with calculators is they rarely have logs other than base e and base 10. So I've got to rely on my conversion being correct to make a graph. Later today I'll use Wolfram Alpha to get accurate numbers and then see what it looks like. I'm back on my laptop now so that should make it tons easier.

Bob

Bob · 2019-10-07 06:54:49

See post 12. I have deleted the graph image here as it isn't correct. I'll leave the function (not correct either) as a lesson in why it's important to check your answers are reasonable. In this case the shape of the graph contradicted the minimum point by calculus so something had to be wrong.

Here's my graph:

DELETED

Function:

(x/1.442695040889)^2 -(2.5694917118487535+1.8204784532536746)x + 4

Bob

Monox D. I-Fly · 2019-10-07 14:25:21

Ah, so indeed there's no solution.

Bob · 2019-10-08 19:41:19

That graph is not consistent with calculus for the minimum point so something is going wrong. Haven't worked out what yet. Assuming it is correct that the graph stays above the x axis, then the expression aY^2 -bY + 4 is always positive for all Y (remember that Y here is the across value not the up). So Y = log(x) > 0 for all x which means that x > 1. So we have a solution. It also seems it doesn't matter which log base we choose, as x > 1 in any base.

I tried the calculations again using Excel with very different results. Maybe I had the format wrong for Wolfram. Anyway it shows the importance of checking whether a result seems reasonable. This time I got a minimum where it should be ie. at Y = 1.52....

So I think This graph is correct:

You can use the quadratic formula to get the exact crossing points and hence the right solution set(s).

Bob

ps. I will amend the earlier post to clarify why that graph image no longer exists.

Math Is Fun Forum

#1 2019-10-03 14:27:17

[ASK]Quadratic Inequation

#2 2019-10-03 19:24:10

Re: [ASK]Quadratic Inequation

#3 2019-10-03 19:57:45

Re: [ASK]Quadratic Inequation

#4 2019-10-04 01:56:33

Re: [ASK]Quadratic Inequation

#5 2019-10-04 14:24:14

Re: [ASK]Quadratic Inequation

#6 2019-10-05 05:18:14

Re: [ASK]Quadratic Inequation

#7 2019-10-05 20:16:15

Re: [ASK]Quadratic Inequation

#8 2019-10-06 14:02:34

Re: [ASK]Quadratic Inequation

#9 2019-10-07 01:49:19

Re: [ASK]Quadratic Inequation

#10 2019-10-07 06:54:49

Re: [ASK]Quadratic Inequation

#11 2019-10-07 14:25:21

Re: [ASK]Quadratic Inequation

#12 2019-10-08 19:41:19

Re: [ASK]Quadratic Inequation

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