Question:
For the equation \(|x^2| + |x| – 6 = 0\), the roots are:
समीकरण \(|x^2| + |x| – 6 = 0\) के लिए, मूल हैं:
Options:
- One and only one real number
- Real with sum one
- Real with sum zero
- Real with product zero
- एक और केवल एक वास्तविक संख्या
- योग एक के साथ वास्तविक
- शून्य योग के साथ वास्तविक
- उत्पाद शून्य के साथ वास्तविक
Correct Answer: c)
Explanation:
Since \(|x^2| = |x|^2\), let \(y = |x| \ge 0\).
The equation becomes:
\[y^2 + y – 6 = 0 \implies (y + 3)(y – 2) = 0\]
Since \(y = |x| \ge 0\), we reject \(y = -3\), so \(y = 2\).
\[|x| = 2 \implies x = 2 \text{ or } x = -2\]
The roots are real numbers \(2\) and \(-2\).
Their sum is \(2 + (-2) = 0\).
Therefore, the roots are real with sum zero.
Hence, the correct option is (c).
माना \(y = |x| \ge 0\)। समीकरण बन जाता है:
\[y^2 + y – 6 = 0 \implies (y + 3)(y – 2) = 0\]
चूँकि \(|x| \ge 0\), अतः \(y = 2 \implies x = \pm 2\)।
मूलों का योगफल:
\[2 + (-2) = 0\]
अतः मूल शून्य योग के साथ वास्तविक हैं।
अतः सही विकल्प (c) है।