A quadratic of the form $-2x^2 + bx + c$ has roots of $x = 3 + \sqrt{5}$ and $x = 3 - \sqrt{5}.$ The graph of $y = -2x^2 + bx + c$ is a parabola. Find the vertex of this parabola.

Guest Apr 23, 2020

#1**+1 **

If a quadratic has roots of 3 + sqrt(5) and 3 - sqrt(5),

the equation can be: ( x - (3 + sqrt(5) )( x - (3 + sqrt(5) ) = 0

Multiplying this out: x^{2} - 6x + 4 = 0 ---> y = x^{2} - 6x + 4

If we want this in the form of y = -2x^{2} + bx + c = 0, we can take our equation and multiply it by -2:

y = -2x^{2} + 12x - 8

You can use a formula to find the vertex, or complete the square:

Completing the square: y = -2x^{2} + 12x - 8

y + 8 = -2x^{2} + 12x

y + 8 = -2(x^{2} - 6x )

y + 8 -18 = -2(x^{2} - 6x + 9)

y - 10 = -2(x - 3)^{2}

The vertex is (3, 10)

geno3141 Apr 23, 2020

#2**0 **

Have you noticed that there are at least 13 posts asking this question? (There are more, but these have answers.)

https://web2.0calc.com/questions/help-im-dying-in-this-problem

https://web2.0calc.com/questions/a-parabola-with-equation-y-ax-2-bx-c-has-a-vertical-line-of-symmetry-at-x-2-and-goes-through-the-two-points-1-1-and-4-1-the-quadratic-ax-2-bx

https://web2.0calc.com/questions/help-me-please_130

https://web2.0calc.com/questions/please-help_11649

https://web2.0calc.com/questions/help-plz_44287

https://web2.0calc.com/questions/need-help-by-the-end-of-today

https://web2.0calc.com/questions/help-plz_5023

https://web2.0calc.com/questions/urgently-need-help-within-two-days

https://web2.0calc.com/questions/quick-math-question_7

https://web2.0calc.com/questions/math-help_2416

https://web2.0calc.com/questions/a-quadratic-of-the-form-2x-2-bx-c-has-roots-of-x

https://web2.0calc.com/questions/help_27016

https://web2.0calc.com/questions/algebra-2-parabolas-please-help

I know you're trying to help, but these people don't even search for existing answers. This shows how little effort they're putting into their math. :-(

As many other questions in this site, this is an AoPS question. AoPS students have a live class, where they learn from an instructor and the class is highly interactive. If you don't understand, then you can ask and there are people ready to help in real time. The online materials for the class are always available for the duration of the course.

Even if you can't make it to the class, you can see what happened later, as there's a transcript of each session. If you're working on your homework and have questions, AoPS also has a forum that is available 24/7. Lots of people are ready to help there. The website is well designed even for use with smartphones, so wherever you go, you get access to all of this. A student can receive hints on how to proceed. When they show some work, they get tailored replies depending on what's the part they're finding hard to understand.

Most students asking AoPS questions here are only looking for a cheap way out.

Guest Apr 26, 2020

#3**0 **

Firstly I will state outright that I have no control over what other people post here.

However. I do ponder a little about your motives.

I am very relieved that I am no longer teaching in a classroom. I expect mobile phones and internet in the classroom would make teaching near impossible at times.

I always wanted my students to learn, that is certainly true. But I cared a lot more about the students that wanted to learn than I did about those that were not interested. Ultimately fully supervised exams would sort out those that cheated on assignments and those who were active learners.

From the little that I know, AoPS appears to be a very reputable internet learning site. Suitable for wealthy families and with scholarships for gifted children. I expect you have a lot of children in your courses who are not gifted and are not interested. I think some of these children are pushed to succeed where they cannot. Hence they cheat.

If your learning aids were of as high a value as you believe them to be then why would so many children come here, either for alternate explanation or for the purpose of cheating.

Melody Apr 26, 2020

#4**0 **

Hi, Melody.

First, I want to say that I believe you have one of the best approaches to this matter in the entire web2.0calc community. Your "Should you consider anything before you answer a question?" is a great guide to answer questions on this forum.

That post above is somehow inspired by that. Some of you know that questions like these get asked multiple times. Many times you guide the asking person to the existing threads on the same question ("See first ..."). I'd say that following those guidelines, you can consider if the question has been asked previously before giving a full answer.

This case is particularly bad. It has lots of answers. Considering your question "Are you seriously trying to help the asker to learn?" on the "Should you..." thread, I think that the student's attitude in this case clearly shows that there's no effort on their part. The community should ponder this before answering the questions.

From the community discussions on the topic of cheating, I've seen that you all have an attitude of wanting to help, which is great. But at the same time I wanted to mention the aids the students have for they are not helpless, which some of you believe they are. If they wanted to take the challenge of these problems, they would use the many resources at their hand, not ask for a full solution to claim it as their own.

You're right that some of these children are pushed to succeed, and that may be one of the major reasons why they cheat. Others are used to cheat. In any case, we can't really do anything to avoid that. But eventually, they'll reap what they've sown.

Thanks.

Guest Apr 26, 2020

#5**0 **

Thanks, AoPS representative,

I wish I had a lot more control here. It makes me sad that this site, which I put so much time into, is largely a cheat site.

**However, I do know that some people come here to learn. Some have learned a tremendous amount here. **

We also provide a free environment where maths lovers of all ages can congregate, participate, and enjoy themselves.

One thing that is important in learning is reinforcement.

If a child comes here to cheat and another child puts a great deal of effort into answering then the second child is learning. I would not want to discourage children from attempting difficult questions. It is a pity that the person on the asking end is cheating but the price is worth it.

When people at a much higher level answer, with full answers, to obvious cheats, it does make me see red.

However, I understand that teaching is a learned art and some of these people have no idea how to teach. They only know how to answer and they want to help. These people are not paid, they do not want to search through to look for old answers, they may not even have the skills to do that well. I usually rely on others if I want to find an old post. It is much easier to ask the members to find it than to look for myself and I figure I am helping them practice a research skill. See, in this example, I am being lazy, the young people help, they practice and reinforce their skill, they are praised, I get the info I want, everyone wins.

I believe I am an excellent one to one, face to face, teacher but I used to get very frustrated in the school classroom because I was expected to teach children material that was far beyond their ability. All this did was make them hate mathematics and since they were being made to feel stupid, they deflected attention by misbehaving. This attracted peer attention and approval so it had a double plus from their point of view.

I do think AoPS must have a great many students in the same category. Many of your students would be there because the parents are pushing them and the parents can afford to pay. I doubt that less intelligent children are able to learn much serious problem-solving mathematics with you, or with anyone else. As for children on scholarships, if they are cheating perhaps it is sometimes because of the amount of pressure that they feel is put on them to succeed. I am not excusing their cheating behaviour, I am just looking at motives.

I am sorry that this site is such a thorn in AoPS's side. As I said, if I had control of the Web2.0 site, I would make some changes.

Melody
Apr 26, 2020