If the two roots of the quadratic equation \(x^2+bx+c=0\) are opposites of each other, which of the following conditions is necessary?
Answer and explanation
Correct answer: \(b=0\)
Let the roots be \(\alpha\) and \(-\alpha\). Their sum is \(\alpha+(-\alpha)=0\). By Vieta’s formula, the sum of the roots of \(x^2+bx+c=0\) is \(-b\). Hence \(-b=0\), so \(b=0\). The condition \(c=0\) is not necessary; for example, \(x^2-1=0\) has roots \(1\) and \(-1\), but \(c=-1\). Exam tip: compare the sum of the roots directly with the coefficient relation \(-b\).
Frequently asked questions
What is the correct answer to this question?
\(b=0\)
Why is this the correct answer?
Let the roots be \(\alpha\) and \(-\alpha\). Their sum is \(\alpha+(-\alpha)=0\). By Vieta’s formula, the sum of the roots of \(x^2+bx+c=0\) is \(-b\). Hence \(-b=0\), so \(b=0\). The condition \(c=0\) is not necessary; for example, \(x^2-1=0\) has roots \(1\) and \(-1\), but \(c=-1\). Exam tip: compare the sum of the roots directly with the coefficient relation \(-b\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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