Find the number of quadratic equations of the form \(x^2 + ax + b = 0,\) such that whenever \(c\) is a root of the equation,\(c^2 - 2\) is also a root of the equation.
+6251 \(x^2+ax+b = (x-c)(x-(c^2-2) = \\ (x-c)(x-c^2+2) = \\ x^2 +(2-c-c^2)x + (c^3-2c) \\\)
\(\text{As we don't have any limitations on }c\\ \text{There are infinitely many equations where both }c \text{ and }c^2-2\\ \text{are both roots, given by }x^2+ax+b \\ a = 2-c-c^2\\b = c^3-2c\)
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+532 this means that a=-c^2-c+2 (vieta's) and b=c^3-2c (also vieta's)
now you can see that all you need to do is plug in any value for c to get any two values for a and b. this means that there are infinitely many solutions to the equation.
+6251 if you're going to waste our time with some trick question please state it ahead of time.
your life may be so invaluable that you have time to waste on b******t, mine isn't.
+118673 Asker guest.
Your response is extremely rude.
If you have reason to think the answers are wrong then you can say so politely and you should give the reason that you think this.
I have looked at Rom's answer and it looks good to me.
Asdf has asked someone to take a look.
Hi asdf,
I do not know what "vieta's" means and I do not understand your response.
I suggest you look at what Rom did.
asdf, when guests gives the response of "this is wrong" you are best to counter with the response "Please explain why you think so".
Encourage the guest to elaborate.
That is if you really care about it. 
+775 Vieta's formulas.
Say we have a polynomial \(ax^2 + bx + c = 0\) with solutions X_1 and X_2.
Vieta's formulas prove that...
a) X_1 + X_2 = -b/a.
b) (X_1) * (X_2) = c/a.
- PM
Giving you the answer? where's the fun in that?
besides, you are downvoting my posts and that is very rude 
+775 You are a guest, so it does not matter. If you have a real accout, then downvoting is truly dissappointing. (*hint hint* i am continuously donvoted by someone)
+532 o.o
because you are saying that us two are wrong when we are right.
unless you mean when c=c^2-2
+532 i upvoted your posts ;)
now pls tell me the answer so i can work backwards
+2489 I suspect this rude, careless dumb-dumb intended this equation:
\(\LARGE x^2 - ax + b = 0\\ \)
Geometric solutions for this quadratic form are found by plotting A(0, 1) and B(a, b) and using the midpoint and distance formulas for finding (y) intercepts in circles. This seems consistent with the question. Examples of these are sometimes included in pre-calculus texts.
The first use of this form comes from Thomas Carlyle (1795–1881), known more for satirical social commentary than mathematics.
GA
+532 Vieta's formulas are formulas that are like shortcuts.
two of them are where you have ax^2+bx+c.
and the sum of roots is -b/a
product is c/a.
+532 vietas is also easy to prove.
lets say you have x^2+bx+c
a=1 for simplicity, it will work for any other case too
so you have (x+j)(x+k).
x^2+(j+k)x+jk.
j and k are the opposites of the roots.
so then -b/a is the sum.
and two negations still have the same product, so it is still c/a.
hope this proof helped!
