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please Help... i dont even know where to start...
#1
x2 + 2x + 3 ≡ (x + a)2 +b
Find the values of the constants a and b
#2
The equation x2 kx + 3 = 0, where k is a constant with no real roots.
find the set of possible values for k, giving your answer in surd form.
#1 isn't bad. Not knowing where to start on it points to some conceptual holes that should probably be filled in by your teacher, since the rest of algebra will only get worse.
Anywho, as an example what if we randomly guess a=1, b=4. [These are really random, not chosen with any foreknowledge]
Then x^2 + 2x + 3 ≡ (x + a)^2 +b = (x+1)^2+4 = x^2+2x+1+4 = x^2+2x+5.
Does x^2+2x+3=x^2+2x+5? No, since 3 != 5, so this guess didn't work. What guesses will work, and when? Can you write algebraic formulas for those conditions? Figure that out and you'll get it.
#2 is more interesting, though. But, if you know the quadratic equation (I assume it's a quadratic and that you just forgot a "+" sign) you can find appropriate k's from the discriminant really quickly. Again this points to some conceptual misunderstandings that probably shouldn't be glossed over.... I could give you the answer, yeah, and how to derive it, but that probably wouldn't help as much as getting someone to explain these things face-to-face with you. Then again if you can figure out the questions from these relatively vague hints you should be fine for the course.
For 2, I assume you mean:
x^2 + kx + 3 = 0
I also assume that the phrase "with no real roots" refers to the equation and not the constant k? (I've never heard of a constant having roots, let alone real roots :p )
Do you know how to solve the following types of equations (quadratic equations) for x?:
ax^2 + bx + c = 0
(For known values a,b,c)
If you do, use that knowledge! Try to write it down for the general case (with a,b,c) and then try it for the case you are given (with x^2 + kx + 3 = 0).
If you don't, I urge you to go back to the basics and understand them!
Heck, I'm still trying to figure out where you got that triple equal sign.
Btw, I was also assuming the triple equals was a typo. If not, then you look to be doing some much more heavy-duty modular arithmetic, but I don't think that's the case. Incidentally #2 becomes a much more interesting question modulo n, maybe with some nice patterns in the solution.
The triple equals sign represents an identity, rather than an equality.