If the equation \(x^2-2\mu x+2\mu=0\) has two real and distinct roots, which condition on \(\mu\) is correct?
Answer and explanation
Correct answer: \(\mu<0\) or \(\mu>2\)
A quadratic equation has two real and distinct roots only when its discriminant satisfies \(D>0\). Here, \(a=1\), \(b=-2\mu\), and \(c=2\mu\), so \(D=b^2-4ac=4\mu^2-8\mu=4\mu(\mu-2)\). Therefore, \(4\mu(\mu-2)>0\), which gives \(\mu<0\) or \(\mu>2\). At \(\mu=0\) or \(\mu=2\), \(D=0\), so the roots are real but equal, not distinct. Exam tip: For two real and unequal roots, always apply the condition \(D>0\).
Frequently asked questions
What is the correct answer to this question?
\(\mu<0\) or \(\mu>2\)
Why is this the correct answer?
A quadratic equation has two real and distinct roots only when its discriminant satisfies \(D>0\). Here, \(a=1\), \(b=-2\mu\), and \(c=2\mu\), so \(D=b^2-4ac=4\mu^2-8\mu=4\mu(\mu-2)\). Therefore, \(4\mu(\mu-2)>0\), which gives \(\mu<0\) or \(\mu>2\). At \(\mu=0\) or \(\mu=2\), \(D=0\), so the roots are real but equal, not distinct. Exam tip: For two real and unequal roots, always apply the condition \(D>0\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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