If the quadratic equation \(x^2-2(a+b)x+(a-b)^2=0\), where a and b are real numbers, has equal roots, which relation holds between a and b?
Answer and explanation
Correct answer: \(ab=0\)
For a quadratic equation \(Ax^2+Bx+C=0\) to have equal roots, its discriminant must be zero: \(D=B^2-4AC=0\). Here, \(A=1\), \(B=-2(a+b)\), and \(C=(a-b)^2\). Thus, \(D=4(a+b)^2-4(a-b)^2=16ab\). Therefore, \(16ab=0\), giving \(ab=0\). Options B and C describe only particular cases, whereas option A gives the necessary and sufficient relation. Exam tip: For equal roots, immediately apply \(D=0\).
Frequently asked questions
What is the correct answer to this question?
\(ab=0\)
Why is this the correct answer?
For a quadratic equation \(Ax^2+Bx+C=0\) to have equal roots, its discriminant must be zero: \(D=B^2-4AC=0\). Here, \(A=1\), \(B=-2(a+b)\), and \(C=(a-b)^2\). Thus, \(D=4(a+b)^2-4(a-b)^2=16ab\). Therefore, \(16ab=0\), giving \(ab=0\). Options B and C describe only particular cases, whereas option A gives the necessary and sufficient relation. Exam tip: For equal roots, immediately apply \(D=0\).
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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