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If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu

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Bunuel
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If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   28 May 2020, 06:01
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If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the value of |m - n| ?

A. 0
B. 1
C. 2
D. 9
E. 18

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E
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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   28 May 2020, 09:35
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Bunuel wrote:
If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the value of |m - n| ?

A. 0
B. 1
C. 2
D. 9
E. 18

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\(x^2 - 7|x| - 18 = 0\)

\(x=0\) is obviously not a solution so \(x\) is either positive or negative

Case 1: When \(x\) is positive (\(|x|=x\)),

\(x^2 - 7|x| - 18 = 0\) becomes \(x^2 - 7x - 18 = 0\)

Or \((x-9)(x+2)=0\) so \(x=9\) or \(-2\)

But since in this case \(x\) is positive, \(x=9\) is only possible root

\(m=9\)

Case 2: When \(x\) is negative (\(|x|=-x\)),

\(x^2 - 7|x| - 18 = 0\) becomes \(x^2 + 7x - 18 = 0\)

Or \((x+9)(x-2)=0\), so \(x=-9\) or \(2\)

But since in this case \(x\) is negative, \(x=-9\) is only possible root

\(n=-9\)

Therefore \(|m-n| = |9-(-9)| = 18\)

Answer is (E)
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If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   Updated on: 28 May 2020, 23:22
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Bunuel wrote:
If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the value of |m - n| ?

A. 0
B. 1
C. 2
D. 9
E. 18




i.e. x^2 - 7|x| - 18 = 0

i.e. lxl^2 - 7|x| - 18 = 0

i.e. |x|(|x|-9)+2(|x|-9)=0

i.e. (|x|-9)(|x|+2)=0

i.e. |x|=9 or -2

But lxl is never negative hence lxl = 9

i.e. x = -9 and +9 which are the values of m and n hence

lm-nl = l-9-9l = 18



Answer: Option E

Originally posted by GMATinsight on 28 May 2020, 06:18.
Last edited by GMATinsight on 28 May 2020, 23:22, edited 1 time in total.
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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   28 May 2020, 06:40
GMATinsight wrote:
Bunuel wrote:
If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the value of |m - n| ?

A. 0
B. 1
C. 2
D. 9
E. 18

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SUm of the roots = -b/a
Product of the roots = c/a

i.e. x^2 - 7|x| - 18 = 0

i.e. Roots = -9 and +9

lm-nl = l-9-9l = 18

Answer: Option E


Sir, shouldn't roots be -9,&2 or -2&9
Because (x+9)(x-9) quadratic equation is x^2-81
Kindly Revert

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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   28 May 2020, 07:06
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\(x^2\)-7|x|-18=0
\(|x|^2\)-7|x|-18=0 --> (\(x^2\) can be written as \(|x|^2\))
\(|x|^2\)-9|x|+2|x|-18=0
|x|(|x|-9)+2(|x|-9)=0
(|x|-9)(|x|+2)=0
|x|=9 or -2

Since modulus value is never negative, discard |x|=-2
Solution of equation:
|x|=9
x=9 or x=-9

Sum of solutions of the above equation = |9-(-9)|=18

Answer: E
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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   28 May 2020, 07:11
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yashikaaggarwal wrote:
GMATinsight wrote:
Bunuel wrote:
If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the value of |m - n| ?

A. 0
B. 1
C. 2
D. 9
E. 18

Project PS Butler


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SUm of the roots = -b/a
Product of the roots = c/a

i.e. x^2 - 7|x| - 18 = 0

i.e. Roots = -9 and +9

lm-nl = l-9-9l = 18

Answer: Option E


Sir, shouldn't roots be -9,&2 or -2&9
Because (x+9)(x-9) quadratic equation is x^2-81
Kindly Revert

Posted from my mobile device


yashikaaggarwal

-2&9 are the values of lxl and not of x

but lxl can not be -ve so -2 is out anyway 2 and -2 do NOT satisfy the equation

so we are left with

lxl = 9

ie.. x = +9 or -9

I hope that help
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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   29 May 2020, 11:18
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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink] New post   10 Dec 2023, 21:11
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m and n are the roots of x^2 - 7|x| - 18 = 0 and we need to find the value of |m - n|



As we have |x| in the equation so we will have two cases
-Case 1: x ≥ 0
=> |x| = x
=> \(x^2 - 7x - 18 = 0\)
=> \(x^2 − 9x + 2x − 18 = 0\)
=> x * ( x-9) + 2*(x - 9) = 0
=> (x - 9) * (x + 2) = 0
=> x = 9 or -2

But condition was x ≥ 0
=> x = 9 is a SOLUTION		-Case 2: x < 0
=> |x| = -x
=> \(x^2 + 7x - 18 = 0\)
=> \(x^2 + 9x - 2x − 18 = 0\)
=> x * ( x+9) - 2*(x + 9) = 0
=> (x + 9) * (x - 2) = 0
=> x = -9 or 2

But condition was x < 0
=> x = -9 is a SOLUTION


=> Both one of m or n is 9 and other is -9
=> |m -n| = |9-(-9)| = |18| = 18 OR
|m -n| = |-9-9| = |-18| = 18 OR

So, Answer will be E
Hope it helps!

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Re: If m and n are the roots of x^2 - 7|x| - 18 = 0, then what is the valu [#permalink]
10 Dec 2023, 21:11
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