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How many solutions are there for this quadratic equation containing modulus?

## Real and imaginary solutions for an equation

How many real solutions are there for the equation x2 – 7|x| - 30 = 0?
1. 3
2. 1
3. 2
4. none

Choice C. 2

## Detailed Solution

Case 1: x > 0
x2 – 7x - 30 = 0
( x - 10 ) ( x - 3 ) = 0
X = 10, x = -3
But x = -3 is not possible as we have considered x > 0, thus 1 solution for this case.
Case 2: x < 0
x2 + 7x - 30 = 0
( x + 10 )( x - 3)=0
X = -10 and x = 3
Only x = -10 is permissible.
Thus this equation has 2 real solutions
Alternatively, we can think of the above as a quadratic in |x|
x2 – 7|x| - 30 = 0 can be factorized as
(|x| -10) ( |x| + 3) = 0
|x| cannot be -3, |x| can only be 10. X can take 2 real values.

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