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I am suppose to find all the solutions to this problem, I think some theorem states that there can only be as many solutions to the problem as the highest degree. I know that calculus reinforces this so I know that

$2x^2 + 4x + 1 = 0$

Can have at most two solutions. In calculus this is proven by the derivative being zero at only somewhere. I can't remember and it isn't important yet.

Anyways I have no idea what to do with this problem. I don't think I can factor it conventionally because of the 2 coefficient so what is the method at this point? I tried guessing and it didn't work at all for -2 - 3.

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Have you tried the quadratic formula...? – icurays1 Dec 2 '12 at 23:53
I do not have the quadratic formula memorized as I don't use it often enough to commit to memory forever. – user138246 Dec 3 '12 at 0:00
Then you need to commit to memory either an alternative solution method, the means to re-derive it when necessary, or where to look it up when necessary. – Hurkyl Dec 4 '12 at 7:28
up vote 1 down vote accepted

Use the Quadratic Formula. One cannot expect all quadratic polynomials to factor "nicely." The Quadratic Formula needs to become a completely standard tool in your arsenal.

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I just don't have the type of memory that can forever remember something if I don't use it often enough. Is there any way to know that i was suppose to use this formula? Why not keep guessing roots? – user138246 Dec 3 '12 at 0:04
The reason that in the past you could guess roots is that the problem-setter chose the coefficients $a$, $b$, and $c$ of $ax^2+bx+c$ so that the roots were easily guessable: either small integers, or, sometimes, rationals with simple denominators. In many of the problems you will meet, the roots are not guessable. If one chooses $a$, $b$, $c$ at random, most of the time the roots are not guessable. – André Nicolas Dec 3 '12 at 0:09

$$ax^2 + bx + c = 0, \text{ roots?}$$
This is where the quadratic formula comes in handy (you should memorize this!):

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

The discriminant d of the quadratic formula is the term $d= (b^2 - 4ac)$. When $b^2\geq 4ac$, you have two real roots.

At the very least, try to memorize how to compute the discriminant. That will allow you to easily determine whether or not a quadratic equation has real roots.

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If you don't want to use the formula, you can do the following.(Actually this is how the formula is derived though. I used to use this method when I didn't memorize the formula.)





$x+1=+1/\sqrt 2$ or $-1/\sqrt 2$

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The quadratic formula works for all quadratic equations. For any equation [in the form] $ax^2 + bx + c = 0$, this is true:$$x = {-b \pm \sqrt{b^2 - 4ac} \over 2a}$$

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You should definitely commit the formula to memory. Here's a cool video to the tune of row-row-row-your-boat, but there are plenty more out there. Once you get on youtube, search around as there are lots of great videos in there that show you how to use the formula.

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