What is the nature of the roots of \(x^2-2(1+\sqrt{5})x+(6+2\sqrt{5})=0\)?
Answer and explanation
Correct answer: Two real and equal \(D=0\)
Here, \(a=1\), \(b=-2(1+\sqrt{5})\), and \(c=6+2\sqrt{5}\). Therefore, the discriminant is \(D=b^2-4ac=4(1+\sqrt{5})^2-4(6+2\sqrt{5})=0\), since \((1+\sqrt{5})^2=6+2\sqrt{5}\). Hence, the roots are real and equal; in fact, the repeated root is \(x=1+\sqrt{5}\). Although this root is irrational, the roots are still equal. Exam tip: determine the nature of the roots from the sign of \(D\) before checking whether the root is rational or irrational.
Frequently asked questions
What is the correct answer to this question?
Two real and equal \(D=0\)
Why is this the correct answer?
Here, \(a=1\), \(b=-2(1+\sqrt{5})\), and \(c=6+2\sqrt{5}\). Therefore, the discriminant is \(D=b^2-4ac=4(1+\sqrt{5})^2-4(6+2\sqrt{5})=0\), since \((1+\sqrt{5})^2=6+2\sqrt{5}\). Hence, the roots are real and equal; in fact, the repeated root is \(x=1+\sqrt{5}\). Although this root is irrational, the roots are still equal. Exam tip: determine the nature of the roots from the sign of \(D\) before checking whether the root is rational or irrational.
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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