Discriminant | Mathematics | Page 1

polar hareBOT Mar 13, 2024, 3:42 AM

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reef onyx Mar 13, 2024, 3:52 AM

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What have you tried?

shadow sorrel Mar 13, 2024, 3:53 AM

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reef onyx What have you tried?

i have no idea how to do it

reef onyx Mar 13, 2024, 3:53 AM

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...you said "discriminant" in the title.

shadow sorrel Mar 13, 2024, 3:53 AM

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reef onyx ...you said "discriminant" in the title.

thats the subtopic of it under the textbook

reef onyx Mar 13, 2024, 3:53 AM

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Okay, so what is the discriminant?

shadow sorrel Mar 13, 2024, 3:53 AM

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b^2-4ac

reef onyx Mar 13, 2024, 3:54 AM

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So in this case?

shadow sorrel Mar 13, 2024, 3:54 AM

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1-8

reef onyx Mar 13, 2024, 3:54 AM

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...no.

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What's a?

shadow sorrel Mar 13, 2024, 3:55 AM

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1^2-4(2)(1)

reef onyx Mar 13, 2024, 3:55 AM

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Still no. Answer the question I asked, please. What's a?

shadow sorrel Mar 13, 2024, 3:55 AM

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2

reef onyx Mar 13, 2024, 3:55 AM

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So what's c?

shadow sorrel Mar 13, 2024, 3:55 AM

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1

reef onyx Mar 13, 2024, 3:56 AM

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shadow sorrel b^2-4ac

Where does this formula come from? What are a, b, and c here? Where do they come from, are they just completely random letters?

shadow sorrel Mar 13, 2024, 3:56 AM

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ax^2+bx+c

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a=2
b=1
c=1

reef onyx Mar 13, 2024, 3:57 AM

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Okay, so then where's your variable b coming in?

shadow sorrel Mar 13, 2024, 3:57 AM

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parabola formula

shadow sorrel Mar 13, 2024, 3:57 AM

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reef onyx Okay, so then where's your variable b coming in?

wdym

frosty raft Mar 13, 2024, 3:57 AM

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use discriminant

reef onyx Mar 13, 2024, 3:57 AM

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What is the question asking?

reef onyx Mar 13, 2024, 3:57 AM

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frosty raft use discriminant

Yes, hi, already helping.

shadow sorrel Mar 13, 2024, 3:57 AM

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find values for b if equation has real roots

reef onyx Mar 13, 2024, 3:58 AM

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What equation?

shadow sorrel Mar 13, 2024, 3:58 AM

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2x^2+x+b+1

reef onyx Mar 13, 2024, 3:58 AM

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So let's make a little change here.

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Let's find the values of y if the equation 2x^2 + x + y + 1 = 0 has real solutions.

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Literally all I did was rename our variable so that maybe it's less confusing.

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So now, in our new equation, what's a?

shadow sorrel Mar 13, 2024, 3:59 AM

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2

reef onyx Mar 13, 2024, 3:59 AM

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And what's b?

shadow sorrel Mar 13, 2024, 3:59 AM

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1

reef onyx Mar 13, 2024, 3:59 AM

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And what's c?

shadow sorrel Mar 13, 2024, 3:59 AM

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1

reef onyx Mar 13, 2024, 3:59 AM

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...but then where's y?

shadow sorrel Mar 13, 2024, 4:00 AM

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in the middle

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idk 💀

reef onyx Mar 13, 2024, 4:00 AM

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Okay, let's maybe go back further to basics.

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Looking at ax^2 + bx + c, a is everything that is multiplied by x^2, b is everything that is multiplied by x, and c is everything else.

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So looking at 2x^2 + x + y + 1, a, b, and c are what?

shadow sorrel Mar 13, 2024, 4:01 AM

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a=2
b=1

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c=y+1

reef onyx Mar 13, 2024, 4:01 AM

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Yes!

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Therefore the discriminant, b^2 - 4ac, is what?

shadow sorrel Mar 13, 2024, 4:02 AM

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2^2-4(1)(y+1)

reef onyx Mar 13, 2024, 4:02 AM

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Almost.

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...no.

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That's worse.

shadow sorrel Mar 13, 2024, 4:02 AM

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wait im slow

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1^2-4(2)(y+1)

reef onyx Mar 13, 2024, 4:02 AM

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Right.

shadow sorrel Mar 13, 2024, 4:02 AM

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💀

reef onyx Mar 13, 2024, 4:03 AM

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And when does the quadratic have real solutions?

shadow sorrel Mar 13, 2024, 4:03 AM

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when > 0

reef onyx Mar 13, 2024, 4:03 AM

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What about when it equals 0?

shadow sorrel Mar 13, 2024, 4:03 AM

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=0

reef onyx Mar 13, 2024, 4:03 AM

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What about when it's less than 0?

shadow sorrel Mar 13, 2024, 4:04 AM

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no

reef onyx Mar 13, 2024, 4:04 AM

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Why not?

shadow sorrel Mar 13, 2024, 4:04 AM

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because its not on x axis

reef onyx Mar 13, 2024, 4:04 AM

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I don't know if that's an answer.

shadow sorrel Mar 13, 2024, 4:04 AM

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because it doesnt intercept the x axis

reef onyx Mar 13, 2024, 4:04 AM

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What's "not on x axis"?

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Okay, but why not?

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Okay, look. Are you familiar with the quadratic formula?

shadow sorrel Mar 13, 2024, 4:05 AM

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-b+-sqrt b^2-4ac/2a

reef onyx Mar 13, 2024, 4:05 AM

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...I'll ignore the notation errors.

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So what's that inside the square root?

shadow sorrel Mar 13, 2024, 4:05 AM

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b^2-4ac

reef onyx Mar 13, 2024, 4:06 AM

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That's our discriminant, right?

shadow sorrel Mar 13, 2024, 4:06 AM

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yes

reef onyx Mar 13, 2024, 4:06 AM

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So then what happens when it's less than 0? When it's negative?

shadow sorrel Mar 13, 2024, 4:06 AM

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parabola concave down

reef onyx Mar 13, 2024, 4:06 AM

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...no.

shadow sorrel Mar 13, 2024, 4:06 AM

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wait

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its not real

reef onyx Mar 13, 2024, 4:06 AM

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Why not?

shadow sorrel Mar 13, 2024, 4:07 AM

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idk

reef onyx Mar 13, 2024, 4:07 AM

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Where did we just find the discriminant?

shadow sorrel Mar 13, 2024, 4:07 AM

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quadratic formula

reef onyx Mar 13, 2024, 4:07 AM

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Where in the quadratic formula?

shadow sorrel Mar 13, 2024, 4:07 AM

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the sqrt

reef onyx Mar 13, 2024, 4:07 AM

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And what happens when you take a square root of a negative number?

shadow sorrel Mar 13, 2024, 4:08 AM

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+-ans

reef onyx Mar 13, 2024, 4:08 AM

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...no.

shadow sorrel Mar 13, 2024, 4:08 AM

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math error 💀

reef onyx Mar 13, 2024, 4:08 AM

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You get an imaginary answer.

shadow sorrel Mar 13, 2024, 4:08 AM

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oh

reef onyx Mar 13, 2024, 4:08 AM

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That's why a negative discriminant gives no real solutions, because it gives complex solutions instead.

shadow sorrel Mar 13, 2024, 4:09 AM

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ok

reef onyx Mar 13, 2024, 4:09 AM

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A complex number, for the record, is just the sum of a real number and an imaginary number.

shadow sorrel Mar 13, 2024, 4:10 AM

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ok

reef onyx Mar 13, 2024, 4:10 AM

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So going way way back to the question, we want to determine when 1^2 - 4 * 2 * (y + 1) >= 0, right?

shadow sorrel Mar 13, 2024, 4:11 AM

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yes

reef onyx Mar 13, 2024, 4:11 AM

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So just... solve the inequality.

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Can you take it from here?

shadow sorrel Mar 13, 2024, 4:11 AM

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1 moment

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1/8 >= b

reef onyx Mar 13, 2024, 4:13 AM

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shadow sorrel 1/8 >= b

Show your work.

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Also, remember, we're trying to solve for y.

shadow sorrel Mar 13, 2024, 4:13 AM

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1^2 - 4(2)(y+1) >=0

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1 - 8y >= 0

reef onyx Mar 13, 2024, 4:14 AM

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Whoa, problem.

shadow sorrel Mar 13, 2024, 4:14 AM

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wait

reef onyx Mar 13, 2024, 4:15 AM

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That was a bunch of steps at once there.

shadow sorrel Mar 13, 2024, 4:15 AM

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1-8y+4

reef onyx Mar 13, 2024, 4:15 AM

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Why?

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One step at a time.

shadow sorrel Mar 13, 2024, 4:15 AM

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8+4y+4

reef onyx Mar 13, 2024, 4:16 AM

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This isn't showing me any work.

shadow sorrel Mar 13, 2024, 4:16 AM

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4x2+4xy+4x1

reef onyx Mar 13, 2024, 4:16 AM

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Stop guessing.

shadow sorrel Mar 13, 2024, 4:16 AM

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ok

reef onyx Mar 13, 2024, 4:16 AM

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Start with 1^2 - 4(2)(y + 1) >= 0.

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And do one thing - exactly one - which you know you are allowed to do by the laws of algebra.

shadow sorrel Mar 13, 2024, 4:17 AM

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(2)(y+1)

reef onyx Mar 13, 2024, 4:17 AM

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And cite the law you're using.

shadow sorrel Mar 13, 2024, 4:17 AM

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distributive law

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2y+2

reef onyx Mar 13, 2024, 4:17 AM

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So what does our inequality read now?

shadow sorrel Mar 13, 2024, 4:18 AM

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1-4(2y+2) >=0

reef onyx Mar 13, 2024, 4:18 AM

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Okay. Now do exactly one thing and cite the law of algebra you're using to justify it.

shadow sorrel Mar 13, 2024, 4:19 AM

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4(2y+2) = 8y+8 distributive law

reef onyx Mar 13, 2024, 4:19 AM

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So what does our inequality read now?

shadow sorrel Mar 13, 2024, 4:19 AM

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1-8y+8 >=0

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9-8y >=0

reef onyx Mar 13, 2024, 4:19 AM

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Wait.

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Stop.

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That's not even an inequality, it's just an expression.

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So where's the minus come from?

shadow sorrel Mar 13, 2024, 4:20 AM

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wait

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i needed to do -4(2y+2)

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right?

reef onyx Mar 13, 2024, 4:20 AM

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I mean, either that or you would have 1^2 - (8y + 8) >= 0.

shadow sorrel Mar 13, 2024, 4:21 AM

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1-8y-8 >=0

reef onyx Mar 13, 2024, 4:21 AM

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Right.

shadow sorrel Mar 13, 2024, 4:22 AM

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-8y-7 >=0

reef onyx Mar 13, 2024, 4:22 AM

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Wait.

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Two steps again.

shadow sorrel Mar 13, 2024, 4:22 AM

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-8+1=-7

reef onyx Mar 13, 2024, 4:22 AM

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8y became positive, why did that happen?

shadow sorrel Mar 13, 2024, 4:23 AM

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reef onyx 8y became positive, why did that happen?

forgor

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-8y-7>=0

reef onyx Mar 13, 2024, 4:23 AM

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Okay.

shadow sorrel Mar 13, 2024, 4:23 AM

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-8y>=7

reef onyx Mar 13, 2024, 4:23 AM

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Right, adding 7 to both sides.

shadow sorrel Mar 13, 2024, 4:23 AM

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mhm

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-8y/-8 7/-8

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y>=7/-8

reef onyx Mar 13, 2024, 4:24 AM

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Wait.

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Something else should happen here.

shadow sorrel Mar 13, 2024, 4:24 AM

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hm

reef onyx Mar 13, 2024, 4:24 AM

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It's an inequality.

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Is 2 > 1?

shadow sorrel Mar 13, 2024, 4:25 AM

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yes

reef onyx Mar 13, 2024, 4:25 AM

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Is -2 > -1?

shadow sorrel Mar 13, 2024, 4:25 AM

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no

reef onyx Mar 13, 2024, 4:25 AM

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So what happened?

shadow sorrel Mar 13, 2024, 4:25 AM

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multipled by -1

reef onyx Mar 13, 2024, 4:25 AM

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And what does that do?

shadow sorrel Mar 13, 2024, 4:25 AM

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changes the equation

reef onyx Mar 13, 2024, 4:26 AM

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...well, it's not an equation, it's an inequality. Changes it how?

shadow sorrel Mar 13, 2024, 4:26 AM

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idk

reef onyx Mar 13, 2024, 4:26 AM

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Is -2 < -1?

shadow sorrel Mar 13, 2024, 4:27 AM

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no

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changes the values

reef onyx Mar 13, 2024, 4:27 AM

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No? So if it's not true that -2 < -1, and it's not true that -2 > -1, then is it true that -2 = -1?

shadow sorrel Mar 13, 2024, 4:28 AM

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wait

shadow sorrel Mar 13, 2024, 4:28 AM

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reef onyx Is `-2 < -1`?

yes

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i read < as >

reef onyx Mar 13, 2024, 4:28 AM

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shadow sorrel yes

So what did we change?

shadow sorrel Mar 13, 2024, 4:29 AM

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the sign

reef onyx Mar 13, 2024, 4:29 AM

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What about the sign?

shadow sorrel Mar 13, 2024, 4:29 AM

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the way its facing

reef onyx Mar 13, 2024, 4:29 AM

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The direction.

shadow sorrel Mar 13, 2024, 4:29 AM

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greater to less than

shadow sorrel Mar 13, 2024, 4:29 AM

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reef onyx The direction.

yes

reef onyx Mar 13, 2024, 4:29 AM

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We changed the direction of inequality.

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We have to do that when we have an inequality and we multiply by a negative or take the reciprocal.

reef onyx Mar 13, 2024, 4:30 AM

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shadow sorrel y>=7/-8

So back here...

shadow sorrel Mar 13, 2024, 4:32 AM

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-8y>=7

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multiply by -1/1?

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idk

reef onyx Mar 13, 2024, 4:34 AM

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...no, we divide by -8. So we...?

shadow sorrel Mar 13, 2024, 4:34 AM

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change the direction

reef onyx Mar 13, 2024, 4:34 AM

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Right.

shadow sorrel Mar 13, 2024, 4:34 AM

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y<=7/-8

reef onyx Mar 13, 2024, 4:34 AM

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Yes.

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Or y <= -7/8.

shadow sorrel Mar 13, 2024, 4:36 AM

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reef onyx Or `y <= -7/8`.

ah

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so for example if it was 8y>=-7

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and we divided both sides by 8

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we would still change the direction of the sign

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or no

reef onyx Mar 13, 2024, 4:41 AM

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shadow sorrel or no

No, because we're not multiplying both sides by a negative number.

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Just a positive one.

shadow sorrel Mar 13, 2024, 4:41 AM

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ok so only if your multiplying by a negative you switch the direction

reef onyx Mar 13, 2024, 4:42 AM

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Or if they're on the same side of 0 and you take the reciprocal.

shadow sorrel Mar 13, 2024, 4:42 AM

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wdym by that

reef onyx Mar 13, 2024, 4:42 AM

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That is to say 0 < a < b implies 0 < 1/b < 1/a.

shadow sorrel Mar 13, 2024, 4:43 AM

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what

reef onyx Mar 13, 2024, 4:44 AM

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Look. 0 < 2 < 3, right?

shadow sorrel Mar 13, 2024, 4:44 AM

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yes

reef onyx Mar 13, 2024, 4:44 AM

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And 0 < 1/3 < 1/2?

shadow sorrel Mar 13, 2024, 4:45 AM

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yes

reef onyx Mar 13, 2024, 4:45 AM

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That's exactly what I'm talking about.

shadow sorrel Mar 13, 2024, 4:46 AM

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ok

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thanks so much for the help <3

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i appreciate it so much

#Discriminant