I need help with the quadratic formula

M

Myrus

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I'm not that experienced in math. In fact, I'm not even in high school. I haven't even really done algebra proper, but I've been messing around with the quadratic formula. I'm not even sure if this should go here in Intermediate Algebra, but it is algebra and is not easy so I put it here.

So, I was trying to derive the quadratic formula using The method of adding b^2, because it's simpler

So I successfully derived it, and then noticed something.

4a^2x^2+4abx+b^2 can be simplified to (2ax+b)^2 or (-2ax-b)^2, since there are 2 answers to square roots either positive or negative.

But with the 2nd simplification, something doesn't work.

(-2ax-b)^2 = b^2-4ac
sqrt
-2ax-b=+or-sqrt(b^2-4ac)
add b
-2ax=b+or-sqrt(b^2-4ac)
/-2a
x=b+or-sqrt(b^2-4ac)/-2a

However, I think is wrong, but, being stupid and bad at math, I can't find my error.

Please help (even though you'll probably find the mistake in 5 seconds).
 
The method of "b^2"? Isn't this usually called "Completing the Square"?

Anyway, \(\displaystyle \dfrac{-b\pm\sqrt{b^{2} - 4ac}}{2a}\;and\;\dfrac{b\pm\sqrt{b^{2} - 4ac}}{-2a}\) are EXACTLY the same. The definition and use of \(\displaystyle \pm\) are a little squishy.
 
GREAT question, but I disagree slightly with tkhunny's answer on this one.

Let's start with a point of logic. "P or Q" is exactly equivalent to "Q or P." Agree?

What does \(\displaystyle x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a}\) mean exactly? It means

\(\displaystyle x = \dfrac{-\ b + \sqrt{b^2 - 4ac}}{2a} \text { OR } x = \dfrac{-\ b - \sqrt{b^2 - 4ac}}{2a}.\)

Now \(\displaystyle x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{(-\ 1)(-\ b \pm \sqrt{b^2 - 4 ac})}{(-\ 1)(2a)} = \dfrac{b \pm (-\ \sqrt{b^2 -4ac})}{-\ 2a}\implies\)

\(\displaystyle x = \dfrac{b + (-\ \sqrt{b^2 - 4 ac})}{-\ 2a} \text { OR } x = \dfrac{b - (-\ \sqrt{b^2 - 4ac})}{-\ 2a} \implies\)

\(\displaystyle x = \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \implies\)

\(\displaystyle x = \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \implies\)

\(\displaystyle x = \dfrac{b \pm \sqrt{b^2 - 4ac}}{-\ 2a}.\)

Nothing squishy about it. It comes from the definition of \(\displaystyle \pm\) as plus OR minus and the reversibility of "or." The two formulas mean EXACTLY THE SAME THING as tkhunny correctly pointed out.
 
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There is no definition of \(\displaystyle \pm\). There is only usage that we all play like we understand. They can't be exactly the same unless it is squishy.

Obviously, opinions vary.

Trouble is, it is only WITH the squishy symbol that the expressions are the same. In order for corresponding solutions to be the same, one must use "+" from one expression and "-" from the other.
 
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