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Algebra. Stuck on question.
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Can someone help me with this question? I'm totally stuck.

The function f is defined by the expression f(x)= 2x+3 ∕ x − 2

For what values on a does the equation f(x) = a − x lack real roots?
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>>45739
Are you trying to isolate a but don't know how?
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>>45743
I'm trying to figure out what values on a make no real roots. That's the premise, and I'm low on intuition.

>x^2 = ax - 2a -3

What values on a will make it so no real roots appear when I apply the pq-formula?
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>>45751
Low on intuition let alone reasoning, hence me asking for help...
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Still in need help. I believe this is the proper channel for math-help requests, as stated in the /sci/ sticky.
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>>45783
in need of* help.
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>>45739
is it
2x + 3 / (x - 2)
or
2x + ( 3 / x ) - 2
>>
The discriminant must be<0, so

a^2-4(2a+3)<0.

You can work from here to deduce things about a
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>>45840
It's "2x + 3 / (x - 2)"
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>>45860
>The discriminant must be<0, so a^2-4(2a+3)<0
Why? Why can't it be >0?

>so a^2-4(2a+3)<0
How did you get this? All I have is x^2 = ax - 2a - 3, and I don't understand how you came to a^2-4(2a+3)<0. I hope me requesting explanations isn't too tedious.
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Since my school doesn't have math teachers I haven't had most of algebra explained/ taught to me. We only get a text book to work through and there is no info on how to solve equations like "x^2 = ax - 2a - 3" in it.
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1)
> op fuck you. it's like 3 am here
2)

2x+3 ∕ x − 2 =a - x

x should be different than 2;(because it's division by 0);
next

2x+3 = ( a - x ) * (x-2) <=>
ax-2a -x^2 +2x -2x -3=0 <=>
-x^2 + ax +2x -2x -2a -3 =0 <=>
-x^2 + ax -2a -3 =0 ;
we need to find the roots where -x^2 =0;
thus delta is
a^2 -4(-1 *(-2a-3)) <=> a^2 - 8a +12 .
for x to not have real we need delta <0
thus a^2 -8a +12 must be < 0.
a^2 -8a +12 =0 <=> delta 2 is
(-8)^2 -4(1*12) = 64 - 48 = 16.
the lowest point is -b/(2a') = 8/2 =4. because a is always positive a^2 -8a +12 cannot be negative... and x will always have solutions. i might have made bad calculations but in theory that is what you have to do.
3
suck my dick i love math.
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it's a^2 - 8a +12 <0 calculate roots. calculate minimum point and then intervals on which a is negative. sorry , i iz sleepy
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>>46096
>It's "2x + 3 / (x - 2)"

>>46100
You made an error in the computation. When you multiply by (x-2) you obtain

2x + 3 / (x - 2) = x - a
2x(x - 2) + 3 = (x - a)(x - 2)
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i think you don't understand some basic algebra here.

imagine the x^2 curve. if you cant, juste type in x^2 on google, you'll get one. all other functions that are written like : ax^2 + bx + c are variants of it, slimmer, larger, turned upside down or not, with their rounded corner going either up or down, left or right in the XY axis, depending on what values a, b, and c are.

Now, the question is, the delta indicates if your x^2 type parabole touches the x-axis line, you can see on wikipedia why, but in delta = b^2 - 4ac , following what i said about how a,b,c influes on the position of your function on the axis, you can easily understand that delta may tell you where your function may be located.

Delta > 0 means it touches the X axis with two branches, therefore you got two ways to touch the x axis with your functions, i.e two solutions

delta = 0 means you only touch the x= 0 axis (or x axis) with the "butt" of the parabole, that is the rounded corner.
delta<0 means the function does not touch the X axis, therefore there is no way oh having f(x) = 0, i.e no solutions.

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>>45739
It's simple, you've already solved most of the problem yourself anon.

You worked out the equation x^2+2a+3-ax=0
Rearrange it as x^2-ax+2a+3 = 0
Now, for exposition ourposes, I'm gonna call a here, say, p.
So let's rewrite it as x^2-px+2p+3

This is a quadratic equation, remember that a quadratic equation is one written in the form ax^2+bx+c, in this case a = 1, b =-p and c=2p+3

Now, the solution to a quadratic equation is given by the quadratic formula, (-b ± sqrt(b^2-4ac))/2a

So, for the values in this equation:

(p ± sqrt(p^2-4(2p + 3))/2

Now, this equation will clearly have a real solution for all values where the square root has a real solution, that is, where p^2-4(2p + 3) >= 0, so it will not have one where p^2-4(2p + 3) < 0

Simplifying the expression we get p^2 -8p -12 < 0

Now, remember that p stood for a in the original question, where it asked to solve find where f(x)= a -x had no real roots.

So, a^2-8a -12 < 0

This gives us two solutions, a< 4 - 2*sqrt(7) and a<4 +2*sqrt(7), or just a <4+2*sqrt(7), as the inequality holds at the middle point of the interval between those two values, a=4.

>>46356
>>a^2 -4(-1 *(-2a-3)) <=> a^2 - 8a +12

The 12 there is negative, everything else that follows is wrong.
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>>46356
>>46393
Can you compress the answer to one sentence? Like what should I write on the paper as the answer?

I don't know what a delta is; isn't mentioned in the math book. Thus, I don't recognize anything below "-x^2 + ax -2a -3 =0 ;" in >>46356
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Anon, >>46356 made a small mistake and the solution isn't correct, I corrected it in >>46458

If you really a need a one line answer, it would be a <4+2*sqrt(7) but I'd strongly encourage you to study the answers you've received so that you can see the reasoning and be able to solve similar problems in the future.
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>>46458
Okay, but if a = 2, then x = 2 ± sqrt 2^2 - 11 is the same as x = 2 ± sqrt (-7), which doesn't work, or something... shouldn't the answer be let's say n>a or n<a... what should replace n then?
...

I fucking hate my school for not having enough math teachers. Our whole fucking country is in lack of math teachers because all we have is goddamn gender studies and quasi-sociologists.
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>>46479
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>>46484
>>46480

Sorry, made a small mistake checking the range, the solution is 4-2*sqrt(7)<a<2+sqrt(7)

Anyway, the key is to practice a lot, and, really, don't rely on teachers, despite what the common word is.Get yourself a good math book at the level you're studying and study from that. This is an elementary algebra problem, some books I could recommend for at this level are "Algebra: a very short introduction" by Peter M. Higgins and "Algebra: A Complete Introduction" by Hugh Neill.
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>>46505
Typed solution wrong, 4-2*sqrt(7)<a<4+2sqrt(7), damnit.
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>>46505
I will check it out. I appreciate you sharing your wisdom. I acknowledge the fact that I have no one to blame but myself for not knowing what I should know, in these times when everything is 'out there'.
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>>46506
See one thing I don't get, and it feels like I'm following you for the most part, is where a< 4 - 2*sqrt(7) comes from?

I have a^2-8a -12, and through the quadratic formula does that not become: a = 4 ± sqrt (4^2+12=28)..? I'm not following.
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>>46587
sqrt(28)=sqrt(4*7)=sqrt(4)*sqrt(7)=2*sqrt(7)
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>>46594
So I can just write 4 - sqrt(28)<a<4 + sqrt(28)?
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>>46612
Yeah.
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>>46618
Thank you so much. You've filled my hole.