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This question and the described solution are copied from a test-paper :

For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.

  1. there is only one root
  2. sum of the roots is + 1
  3. sum of the roots is zero
  4. the product of the roots is +4

Answer : (3)

Answer Explanation :

If x > 0 |x| = x.

Given equation will be $x^2 + x - 6 = 0$⇒ x = 2,- 3 ⇒ x = 2

If x < 0 |x| = - x.

Given equation will b e $x^2$ - x - 6 = 0 ⇒ x = -2, 3 ⇒x = - 2

Sum of roots is 2 - 2 = 0

Now I have a doubt on the statment "If x < 0 |x| = - x." I think modulus means that |x| is always positive ?! Also I can see that (2) seems to be the correct option isn't ?!

Please post your views.

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    It's definitely not an uncommon situation, @Arturo. :)2010-10-21

4 Answers 4

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$\rm\ 0\ =\ x^2 + |x| -6\ =\ (|x| - 2)\:(|x| + 3)\ \Rightarrow\ |x| = 2\ \Rightarrow\ x = \pm2 $

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$f(x)=x^2+|x|-6$ is an even function—that is, $f(x)=f(-x)$ for all $x$, or the graph of $y=f(x)$ is symmetric over the y-axis—so if $f(c)=0$ then $f(-c)=0$, so the sum of the zeros of $f$ must be 0.

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    Yes, or because $f$ is a polynomial function on $(0,\infty)$ and on $(-\infty,0)$. I was just having fun, but also pointing out that technically *something* about the function must be used beside the fact that it is even.2010-10-21
2

If x<0, then -x>0, which means that the modulus is indeed positive.

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    @Ross Millikan :Aha! that's a nice explanation Ross :) Really appreciated :)2010-10-21
2

The statement is saying:

$|x| = -x$ (for $x < 0$)

and divide both sides by $x$ to give us:

$\frac{|x|}{x} = -1$

Let us test a few values to make sure this holds.

$x = -3$ and $|x| = 3$:

$\frac{|-3|}{-3} = -1$

$\frac{3}{-3} = -1$

$-1 = -1$

True.

$x = -5$ and $|x| = 5$

$\frac{5}{-5} = -1$

$-1 = -1$

True.

We can verify this for all $x < 0$.

If you don't believe me, here is a plot of $y = \frac{|x|}{x}$.

plot1


Alternative:

If we define

-a is the number such that a + (-a) = 0 

ie,

if a = 3, -a = -3, because $3 + (-3) = 0$

if a = -2, -a = 2, because $-2 + 2 = 0$

We can call 3 and -3 "opposites".

The opposite of -2 is 2.

The opposite of 5 is -5.

Then we can translate the original statement then as:

If x < 0, |x| is the opposite of x 

Let's test this out.

$x = -4$

$|x| = 4$

$-4 + 4 = 0$

$|-4|$ (or, 4) is indeed the opposite of $-4$.

More generally, the statement says that, if $x < 0$,

$x + |x| = 0$

Which we can verify as true for all values $x < 0$

If you don't believe me, you can look at the plot of $y = x + |x|$

plot2

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    I also noted that I completely missed the point of the question.2010-10-21