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I am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a question - for example 2x^2+7x-15=0 - can I use the simultaneous equation for this or do I just use the factorizing method?

Harry Kitchener
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  • I am not sure what the simultaneous equation you speak of is, but I think it is safe to say that if you already know the quadratic formula, it is safe to use at your discretion on any quadratic. However, if you have only just learned it recently, as well as these other methods (factoring, completion of the square, etc.) then your teacher may request you use one specific method on assignments or exam problems, to test your knowledge of those techniques, in which case you should of course use those. – izœc Apr 03 '14 at 16:11
  • (cont.) More generally, the formula will work for any quadratic-however, for simple ones, factoring might often be more direct (efficient). – izœc Apr 03 '14 at 16:11
  • Thanks, also can I place the factors of 2(-15) = 30 in any order eg:

    (2x^2+10x)(-3x-15) or (2x^2-3x)(10x-15)

     – Harry Kitchener Apr 03 '14 at 16:19
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    Well, the order does matter. If you meant $2x^2 + 10x - (3x + 15) = 2x(x+5) -3 (x+5) = (x+5)(2x-3)$ then it works, but if you try it the other way it doesn't quite work out as directly. I think Allesandro's answer illustrates this well. – izœc Apr 03 '14 at 16:43

4 Answers4

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I always try to use the factorizing method first. But sometimes the method does not work for certain quadratic equations. In that case, I resort to using the quadratic formula $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ For your equation of $2x^2+7x-15=0$, we have $a=2,b=7,c=-15$.

user140170
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If you can factorize your quadratic without using the formula then you should do it, because it is usually faster.

When you have a quadratic $ax^2+bx+c$ you can easily factorize it if you can find two numbers $n_1,n_2$ such that $n_1+n_2=b$ and $n_1n_2=ac$ by rewriting $bx$ as $n_1x+n_2x$ and then grouping similar terms.

In the case of $2x^2+7x-15=0$ we have $n_1=10$ and $n_2=-3$ so we factorize it without having to use the quadratic formula:

$2x^2+7x-15=0$
$2x^2+10x-3x-15=0$
$2x(x+5)-3(x+5)=0$
$(2x-3)(x+5)=0$

Alessandro Codenotti
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  • How would I know which side to factorise the 10x and the 3x on? – Harry Kitchener Apr 03 '14 at 17:58
  • You have to factorize so to have a common term that you can factorize again in the next step (in this case the common factor is $x-5$), there isn't a general rule, but usually you can just "see" it (especially after having solved similar excercises) – Alessandro Codenotti Apr 03 '14 at 18:09
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Key Idea $\ $ Use AC-method to reduce to factoring a polynomial that's $\,\rm\color{#c00}{monic}\,$ (lead coeff $=\color{#c00}1)$

$$\quad\ \ \begin{eqnarray} f &\,=\,& \ \ 2\ x^2+\ 7\ x\ -\,\ 15\\ \Rightarrow\ 2f &\,=\,&\ (2x)^2\! +7(2x)-30\\ &\,=\,& \ \ \ \color{#c00}{X^2+\,7\ X\ -\,\ 30},\,\ \ X\, =\, 2x\\ &\,=\,& \ \ \,(X-3)\ (X+\,10)\\ &\,=\,& \ \ (2x-3)\,(2x+10)\\ \Rightarrow\ f\,=\, 2^{-1}(2f) &\,=\,& \ \ (2x - 3)\,(x+5)\\ \end{eqnarray}\qquad\qquad$$

Community
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Bill Dubuque
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It's my view that the quadratic formula should always be used to factor a quadratic. That's what it's for.

Completing the square should be used to transform a quadratic into vertex-focus form (and to derive the quadratic formula).

Factoring by grouping is only really taught at the HS level to give students an introduction to Number Theory (finding integer solutions to equations, analyzing factors and residuals, etc), or perhaps as a preliminary for developing strategies for factoring cubics. Other than that, it's not very useful.

John Joy
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