why is discriminant greater than or equal to zero if a given quadratic equation f(x) has real x

[fallen drift]

this is something we have to do while finding range

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[past onyx]

You can prove it by just deriving the quadratic formula for ax^2 + bx + c = 0.

[shadow root]

It's more that discriminant is greater than or equal to 0 if f(x)=0 has one or more real solutions

[shadow root]

And yea try looking at the quadratic formula to see why

[steady pilot]

I mean, defining what the discriminant is within the quadratic formula is also a start. Then notice what happens if, say, it is 0, or less than 0.

[fallen drift]

I understand that if discriminant is greater than or equal to zero then roots are real. But on what base do we conclude that roots of f(X)are real given that x is real

[steady pilot]

“x is real”? What does this mean?

[fallen drift]

when this kind of question comes we start it with discriminant greater than or equal to 0 ie D>=0

find the range of f(x)=y= x^2-3x+2

where does the inequality of d come from in these questions

[steady pilot]

It gives the y-coordinate of the vertex?

Not entirely sure what you mean here

[sinful quest]

Think about the quadratic formula

Particularly the square root part

[shadow root]

We conclude that the there are two roots of f(x) are real if b^2-4ac>0

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

[sharp spruce]

Muus

[shadow root]

Looking at the quadratic formula we can see why

We notice the discriminant is the term inside the square root

If b^2-4ac is greater than 0

Then because of the plus minus we get two solutions

If b^2-4ac=0, then we'll get repeated roots as

$\frac{-b+0}{2a}=\frac{-b-0}{2a}$

[sharp spruce]

Muus

[shadow root]

So both roots are equal so really there's only one real root

Then it b^2-4ac is less than 0, then the square root will give us an imaginary number

So no real solutions

[steady pilot]

@shadow root Respectfully, they already said they understand it.

[shadow root]

Welp rip

[steady pilot]

I understand that if the discriminant is greater than or equal to zero then roots are real.

[shadow root]

I was answering the based on what do we conclude

Part of the question

[fallen drift]

Thanks to all of you for responding but I am not talking about the nature of roots based on discriminant. My question is on a different topic: Finding range of a quadratic/quadratic function.

This

Now, I want to ask why did we put D >0 or D=0here

[past onyx]

Not really sure what's going on here, to be honest.

I'd start by simplifying the fraction.

[fallen drift]

in (i) : we take x^2+1 to the left side to multiply by y and then we form a new quadratic equation in x

[past onyx]

Not really sure what the point of doing that is, to be honest.

You can do the following:
f(x) = (x^2 + x + 1)/(x^2 + 1) = 1 + x/(x^2 + 1)
We can then look at g(x) = f(x) - 1 = x/(x^2 + 1). It's an odd function with g(0) = g(+∞) = 0, so its range is easy to find.

And a similar approach for (ii).

[fallen drift]

Well this approach somehow worked in this question, but I don't believe it would in questions of quadratic/quadratic where you can not find the roots

This implies that the discriminant is greater than or equal to zero but why

[paper estuary]

display_name1233 has been timed out for 5m
Reason: Similar-Messages Spam

display_name1233 | 1199995427676753970

[steady pilot]

What the fuck did you do to timeout yourself?

[steady pilot]

Because all outputs in x have to be real in the equation.

I think I see what you’re talking about now.

[fallen drift]

bro i did nothing, it says similar messages scam. Maybe because I have been trying to ask the same thing again and again.

[steady pilot]

Lel

Anyways, that was besides the point

[fallen drift]

so this means that because all values of x are real, whatever the output is it will be real too?

[steady pilot]

What I mean by this is that since we are technically considering a quadratic in x, with y as a constant, we must have x real and therefore the discriminant >=0.

Try to consider it that way.

[fallen drift]

ohh

yess i understand now

thank you @steady pilot

how do i close this now?

[steady pilot]

You’re welcome. Thanks for giving the example, it cleared it up.

The command is +close

[fallen drift]

ofcourse

+close

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