If \(p(x)=x^2-2\sqrt{5}x+5\), which statement about its zeroes is correct?
Answer and explanation
Correct answer: Both zeroes are \(\sqrt{5}\)
The quadratic factors as \(x^2-2\sqrt{5}x+5=(x-\sqrt{5})^2\). Hence the root \(\sqrt{5}\) has multiplicity two and both zeroes equal \(\sqrt{5}\). The closest distractor (option C) is wrong because the polynomial does not have two distinct roots; option D is wrong because the discriminant is zero, so a real repeated root exists. Exam tip: compute the discriminant \(b^2-4ac\); if it equals zero, the quadratic has a repeated real root.
Frequently asked questions
What is the correct answer to this question?
Both zeroes are \(\sqrt{5}\)
Why is this the correct answer?
The quadratic factors as \(x^2-2\sqrt{5}x+5=(x-\sqrt{5})^2\). Hence the root \(\sqrt{5}\) has multiplicity two and both zeroes equal \(\sqrt{5}\). The closest distractor (option C) is wrong because the polynomial does not have two distinct roots; option D is wrong because the discriminant is zero, so a real repeated root exists. Exam tip: compute the discriminant \(b^2-4ac\); if it equals zero, the quadratic has a repeated real root.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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