What is the nature of the roots of the equation \(x^2-2(2+\sqrt{3})x+(7+4\sqrt{3})=0\)?
Answer and explanation
Correct answer: Two real and equal
Here, \(a=1\), \(b=-2(2+\sqrt{3})\), and \(c=7+4\sqrt{3}\). Therefore, the discriminant is \(D=b^2-4ac=4(2+\sqrt{3})^2-4(7+4\sqrt{3})=0\). Hence, the two roots are real and equal. In fact, the repeated root is \(x=2+\sqrt{3}\). Option D is incorrect because the root is irrational but not distinct. Exam tip: When \(D=0\), a quadratic equation has two real and equal roots.
Frequently asked questions
What is the correct answer to this question?
Two real and equal
Why is this the correct answer?
Here, \(a=1\), \(b=-2(2+\sqrt{3})\), and \(c=7+4\sqrt{3}\). Therefore, the discriminant is \(D=b^2-4ac=4(2+\sqrt{3})^2-4(7+4\sqrt{3})=0\). Hence, the two roots are real and equal. In fact, the repeated root is \(x=2+\sqrt{3}\). Option D is incorrect because the root is irrational but not distinct. Exam tip: When \(D=0\), a quadratic equation has two real and equal roots.
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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