Which of the following equations has real, irrational, and distinct roots?
Answer and explanation
Correct answer: \(x^2-2\sqrt{2}x-1=0\)
For option A, the discriminant is \(D=b^2-4ac=(-2\sqrt{2})^2-4(1)(-1)=8+4=12\). Since \(D>0\), the roots are real and distinct. Also, \(\sqrt{D}=\sqrt{12}=2\sqrt{3}\) is irrational, giving the roots \(\sqrt{2}+\sqrt{3}\) and \(\sqrt{2}-\sqrt{3}\), both of which are irrational. In option B, the discriminant is zero; option C has a negative discriminant; and option D has the rational roots 2 and 3. Exam tip: real and distinct roots require \(D>0\), while irrational roots require the square root of the discriminant to be irrational.
Frequently asked questions
What is the correct answer to this question?
\(x^2-2\sqrt{2}x-1=0\)
Why is this the correct answer?
For option A, the discriminant is \(D=b^2-4ac=(-2\sqrt{2})^2-4(1)(-1)=8+4=12\). Since \(D>0\), the roots are real and distinct. Also, \(\sqrt{D}=\sqrt{12}=2\sqrt{3}\) is irrational, giving the roots \(\sqrt{2}+\sqrt{3}\) and \(\sqrt{2}-\sqrt{3}\), both of which are irrational. In option B, the discriminant is zero; option C has a negative discriminant; and option D has the rational roots 2 and 3. Exam tip: real and distinct roots require \(D>0\), while irrational roots require the square root of the discriminant to be 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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