What is the condition on p for the equation \(x^2+2px+49=0\) to have real roots?
Answer and explanation
Correct answer: \(p\leq -7\) or \(p\geq 7\)
For real roots the discriminant must satisfy \(D\ge0\). Here \(D=(2p)^2-4\cdot1\cdot49=4p^2-196=4(p^2-49)\). Requiring \(D\ge0\) gives \(p^2-49\ge0\) or \(p^2\ge49\), hence \(p\le-7\) or \(p\ge7\). Option B is incorrect because for \(-7<p<7\) we have \(p^2<49\) so \(D<0\); option C is just a special value and not the general condition; option D is too broad and includes values that do not give real roots. Exam tip: For parameter problems, compute \(D=b^2-4ac\) first and set \(D\ge0\) to find the range of the parameter.
Frequently asked questions
What is the correct answer to this question?
\(p\leq -7\) or \(p\geq 7\)
Why is this the correct answer?
For real roots the discriminant must satisfy \(D\ge0\). Here \(D=(2p)^2-4\cdot1\cdot49=4p^2-196=4(p^2-49)\). Requiring \(D\ge0\) gives \(p^2-49\ge0\) or \(p^2\ge49\), hence \(p\le-7\) or \(p\ge7\). Option B is incorrect because for \(-7<p<7\) we have \(p^2<49\) so \(D<0\); option C is just a special value and not the general condition; option D is too broad and includes values that do not give real roots. Exam tip: For parameter problems, compute \(D=b^2-4ac\) first and set \(D\ge0\) to find the range of the parameter.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Introduction to Quadratic Equations.
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