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December 17th, 2015, 10:43 AM   #1
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Absolute value term in a quadratic.

Example: x^2 + |x - 7| - 9 = 0

Can be solved this way :
x^2 + SQRT[(x - 7)^2] - 9 = 0
SQRT[(x - 7)^2] = 9 - x^2
squaring both sides and simplifying:
x^4 - 19x^2 + 14x + 32 = 0

Soooo....got rid of absolute term |x - 7|

After googling "quadratic with absolute value term",
I saw a few ways to handle these, but not the above.
Did I simply "invent something stupid"?
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December 17th, 2015, 11:02 AM   #2
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That looks fairly normal to me.

I'd consider writing $y=x-7$ so that $x^2=y^2+14y+49 = |y|^2\pm 14|y|+49 $ and substituting into the original equation allows the use of the quadratic formula to garner solutions.

Last edited by v8archie; December 17th, 2015 at 11:48 AM.
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December 17th, 2015, 02:45 PM   #3
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You have two possibilities: x^2+x-16 (x > 7) or x^2-x-2 (x < 7). Solve for x in each case and use any solution fitting the condition.

Note: I found no solution for the first case and 2 for the second.

Last edited by mathman; December 17th, 2015 at 11:45 PM. Reason: correct error
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December 17th, 2015, 03:11 PM   #4
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I do think that is probably the easiest approach.
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December 17th, 2015, 05:22 PM   #5
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Quote:
Originally Posted by mathman View Post
You have two possibilities: x^2+x-16 (x > 7) or x^2-x-2 (x < 7).
Ya, that's the "popular" approach; I was finding a one operation
to replace the two operations...no big deal...
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December 18th, 2015, 03:47 PM   #6
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Quote:
Originally Posted by Denis View Post
Ya, that's the "popular" approach; I was finding a one operation
to replace the two operations...no big deal...
The x > 7 branch has no roots - by inspection, since the polynomial is positive and increasing for that case. This leaves a simple quadratic for x < 7.
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December 18th, 2015, 04:29 PM   #7
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Once more: I simply made up an equation.
Purpose was not to solve it, but use it as illustration.
Merci.
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December 19th, 2015, 04:08 PM   #8
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Quote:
Originally Posted by Denis View Post
Once more: I simply made up an equation.
Purpose was not to solve it, but use it as illustration.
Merci.
You replaced a quadratic equation (two equations actually), easy to handle, into a fourth degree equation. In general - not a good idea.
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December 19th, 2015, 05:18 PM   #9
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Agree...but much easier when writing a program...my only purpose.

That one was easy to handle, but not all are...
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