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Let D be the absolute value of the difference of the 2 roots of the equation 3x^2-10x-201=0. Find [D]. [x] denotes the greatest integer less than or equal to x.

I came across this question in a Math Competition and I am not sure how to solve it without using a calculator, since calculators are not allowed in the competition. Thanks.

snivysteel
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3 Answers3

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Hint: If $a, b$ are the roots, $$|a-b|^2 = (a-b)^2 = (a+b)^2 - 4 ab = \frac{100}9+4\times \frac{201}3 = \frac{2512}9$$

Macavity
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  • I've never seen this relationship between the roots of a quadratic... Is this a serious gap in my education, or a more minor note that I missed along the way? Also, I can't seem to see the relationship between this and the original quadratic... – abiessu Jun 03 '14 at 12:29
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    @abiessu $|a-b|^2=(a-b)^2=(a+b)^2-4ab$ holds true for all real $a, b$, not just roots. We use the fact that they are roots to quickly find the values of $a+b, ab$ using coefficients of the quadratic (see http://en.wikipedia.org/wiki/Quadratic_equation#Vieta.27s_formulas). Useful to know, IMHO. – Macavity Jun 03 '14 at 12:33
  • Got it, it just took me an extra moment to realize where the formulas came from. – abiessu Jun 03 '14 at 12:56
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    @abiessu The possible gap is the Fundamental Theorem of Symmetric Polynomials - see my answer. The essence of the matter will become clearer if you study Galois Theory. – Bill Dubuque Jun 03 '14 at 15:15
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Hint $\ $ By $ $ Vieta, $\,\ x^2 -\frac{10}3 x - 67\, =\, (x-a)(x-b)\iff \ \color{#0a0}{a+b} = 10/3,\ \color{#c00}{ab} = -67$

$(a-b)^2$ is symmetric in $\,a,b\,$ so by FTSP it can be written as a polynomial in $\,\color{#0a0}{a+b},\ \color{#c00}{ab}$

Indeed, applying Gauss's Algorithm we find that $\, (a-b)^2 = (\color{#0a0}{a+b})^2 -4\color{#c00}{ab}\, =\, \dfrac{16\cdot 157}9$

Remark $\ $ The same algorithm works for polynomials in any number of variables. It reduces problem like this to rote mechanical computation, i.e. no guesswork is required to solve such problems, only simple polynomial arithmetic. The algorithm yields a constructive interpretation of the FTSP = Fundamental Theorem of Symmetric Polynomials, that every symmetric polynomial has a unique representation as a polynomial in the elementary symmetric polynomials.

Gauss's algorithm may be viewed as a special case of Gröbner basis methods (which may be viewed both as a multivariate generalization of the (Euclidean) polynomial division algorithm, as well as a nonlinear genralization of Gaussian elimination for linear systems of equation). Gauss's algorithm is the earliest known use of such a lexicographic order for term-rewriting (now mechanized by the Grobner basis algorithm and related methods).

Community
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Bill Dubuque
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  • Very nice; I had missed symmetric polynomials in my undergraduate education, and while I have noticed them on occasion here on Math.SE and elsewhere, I haven't had much chance to incorporate them into what I understand of mathematics. – abiessu Jun 03 '14 at 16:04
  • @abiessu Symmetry plays a large role in solving many problems, so it is well-worth studying it (one will learn the basics in most courses on abstract agebra). Above it is a bit overkill to use Gauss's algorithm, but I wanted to make it clear that the process is completely mechanical/algorithmic - nothing needs to be pulled out of a hat like magic. Here are some other answers that exploit innate symmetry. – Bill Dubuque Jun 03 '14 at 16:11
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$$3x^2-10x-201=0\\ \iff x^2-\frac{10}3x-67=0$$ Assuming the quadratic formula is available to use, $$x=\frac{10}6\pm\frac{\sqrt{\frac{100}9+4\cdot 67}}2\\=\frac 53\pm\sqrt{\frac{25}9+67}$$

So the square root term is greater than $\sqrt{64}$ but less than $\sqrt{81}$ and is therefore between $8$ and $9$ in value, and therefore the difference between the roots can be either be $16$ or $17$, but not $18$ since that term is less than $9$.

The difference can be determined this way because both roots have the offset fraction $\dfrac53$ which is removed upon subtraction.

To determine whether the difference is greater than $17$, consider whether the square root term is greater than $8.5:$

$$(8+0.5)^2=64+2\cdot8\cdot0.5+0.25=67+5+0.25$$

And our original square root term contains

$$67+\frac{25}9=67+2+\frac79$$

Therefore, half of the difference between the roots is less than $8.5$ but greater than $8$ and therefore the total difference is between $16$ and $17$, leaving $16$ as the value of $[D]$.

abiessu
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  • This is a "brute force" way of attacking this problem, Macavity has listed a good shortcut in another answer. – abiessu Jun 03 '14 at 12:32