What is the nature of the roots of the quadratic equation \(x^2+2(2-\sqrt{5})x+9=0\)?
Answer and explanation
Correct answer: No real roots
Here, \(a=1\), \(b=2(2-\sqrt{5})\), and \(c=9\). Therefore, the discriminant is \(D=b^2-4ac=4(2-\sqrt{5})^2-36=4(9-4\sqrt{5})-36=-16\sqrt{5}<0\). Since \(D<0\), the equation has no real roots. Option B is incorrect because two equal real roots require \(D=0\). Exam tip: for a quadratic equation, \(D<0\) indicates that the roots are non-real.
Frequently asked questions
What is the correct answer to this question?
No real roots
Why is this the correct answer?
Here, \(a=1\), \(b=2(2-\sqrt{5})\), and \(c=9\). Therefore, the discriminant is \(D=b^2-4ac=4(2-\sqrt{5})^2-36=4(9-4\sqrt{5})-36=-16\sqrt{5}<0\). Since \(D<0\), the equation has no real roots. Option B is incorrect because two equal real roots require \(D=0\). Exam tip: for a quadratic equation, \(D<0\) indicates that the roots are non-real.
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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