If \(a\) and \(b\) are real numbers, which of the following relations is necessary for the equation \(x^2-2(a+b)x+(a^2+b^2)=0\) to have real roots?
Answer and explanation
Correct answer: \(ab\geq 0\)
The discriminant is \(D=[-2(a+b)]^2-4(a^2+b^2)=4(a+b)^2-4(a^2+b^2)=8ab\). For real roots, \(D\geq0\), so \(8ab\geq0\), which gives \(ab\geq0\). Option B generally gives \(D\leq0\) and does not ensure real roots. Exam tip: For a quadratic equation, first apply the condition \(D\geq0\) to test for real roots.
Frequently asked questions
What is the correct answer to this question?
\(ab\geq 0\)
Why is this the correct answer?
The discriminant is \(D=[-2(a+b)]^2-4(a^2+b^2)=4(a+b)^2-4(a^2+b^2)=8ab\). For real roots, \(D\geq0\), so \(8ab\geq0\), which gives \(ab\geq0\). Option B generally gives \(D\leq0\) and does not ensure real roots. Exam tip: For a quadratic equation, first apply the condition \(D\geq0\) to test for real 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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