Submitted by Hypatia on Fri, 01/13/2017 - 09:31
Forums:
This is possibly one of the "worst" quadratics to try and solve:
\[
x^2 + ax + b = c \, .
\]
I'll leave it here for a week and then post the solutions.
Submitted by Hypatia on Fri, 01/13/2017 - 09:31
This is possibly one of the "worst" quadratics to try and solve:
\[
x^2 + ax + b = c \, .
\]
I'll leave it here for a week and then post the solutions.
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Am I missing something?
I just plugged the coefficients into the quadratic formula and got \[x=\frac{-a\pm\sqrt{a^2+4c-4b}}{2}\] That didn't seem too bad. Am I missing the point?
Labels
It's not too bad, as long as you keep track of your labels! It's a very STEP-ish thing to do, using labels in places that you would not usually expect them (so the coefficient of $x$ is $a$ rather than $b$ as you might expect). Other examples include sequences where the common ratio is $a$, or the common difference is $r$ (which might trick you into using a geometric formulae).
What usually trips people up (and is why this sort of quadratic is usually harder to solve than others) is when they write out $x=\dfrac {-b \pm \sqrt{b^2-4ac}} {2a}$ and then $b=a$, etc. It is then very easy to make a mistake such as making the denominator equal to $2b$ (from using $b=a$ "incorrectly").
If you have one of these types of question, the best thing is to change the labels in the standard formulae before starting. For example, using $u_n=A + nD$, or for a quadratic:
\[
x=\dfrac {-B \pm \sqrt{B^2-4AC}} {2A} \quad \text{or} \quad x=\dfrac {-b' \pm \sqrt{b'^2-4a'c'}} {2a'}
\]
You can then write "$B=a"$ etc.
Answer
I agree with your answer :-)
I see
I see what you mean. I actually did it by completing the square at first, thinking that something useful might happen, but when I looked at my answer, I realised I could have just plugged it into the formula. I know that's always the case, but generally I prefer completing the square for integer coefficients - it seems to result in fewer mistakes.