For a real parameter \(n\), which statement correctly describes the nature of the roots of the equation \(x^2+2(n+2)x+n^2+4n+8=0\)?
Answer and explanation
Correct answer: No real roots
Here, \(a=1\), \(b=2(n+2)\), and \(c=n^2+4n+8\). Therefore, the discriminant is \(D=b^2-4ac=[2(n+2)]^2-4(n^2+4n+8)=-16\). Since \(D<0\), the equation has no real roots for any real value of \(n\); its roots are complex. Option B would require \(D=0\), while option C would require \(D>0\). Exam tip: determine the nature of quadratic roots from the sign of the discriminant.
Frequently asked questions
What is the correct answer to this question?
No real roots
Why is this the correct answer?
Here, \(a=1\), \(b=2(n+2)\), and \(c=n^2+4n+8\). Therefore, the discriminant is \(D=b^2-4ac=[2(n+2)]^2-4(n^2+4n+8)=-16\). Since \(D<0\), the equation has no real roots for any real value of \(n\); its roots are complex. Option B would require \(D=0\), while option C would require \(D>0\). Exam tip: determine the nature of quadratic roots from the sign of the discriminant.
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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