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Question
Write the number of real roots of the equation x2 + 3 |x| + 2 = 0.
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Solution
Key concept |x| = -x for x < 0
|x| = +x for x > 0
For case I: x < 0
⇒ x2 - 3x + 2 = 0
⇒ x2 - (2 + 1)x + 2 = 0
⇒ x2 - 2x - x + 2 = 0
⇒ x(x - 2) -1 (x - 2) = 0
⇒ (x - 1) (x - 2) = 0
⇒ x = 1, 2
But for |x| = -x i.e x < 0
Here value of x = 1, 2 is > 0
So, condition does not satisfy.
For x > 0 i.e |x| = +x
x2 + 3x + 2 = 0
⇒ x2 + (2 + 1)x + 2 = 0
⇒ x2 + 2x + x + 2 = 0
⇒ x (x + 2) +1 (x + 2) = 0
⇒ (x + 1) (x + 2) = 0
⇒ [x = -1, -2]
Again x < 0 here and we know that |x| = +x for x > 0
So, condition does not satisfy.
Therefore, No real roots exist.
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