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If the expression \(x^2+px+q\) can be written as \((x+r)^2\) for some real number \(r\), which relation between \(p\) and \(q\) is necessary and sufficient?
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Answer and explanation
Correct answer: \(p^2=4q\)
Expanding \((x+r)^2\) gives \(x^2+2rx+r^2\), so \(p=2r\) and \(q=r^2\). Hence \(p^2=4r^2=4q\). The other relations do not ensure a perfect square. Exam tip: square the middle coefficient and compare it with four times the constant term.
Frequently asked questions
What is the correct answer to this question?
\(p^2=4q\)
Why is this the correct answer?
Expanding \((x+r)^2\) gives \(x^2+2rx+r^2\), so \(p=2r\) and \(q=r^2\). Hence \(p^2=4r^2=4q\). The other relations do not ensure a perfect square. Exam tip: square the middle coefficient and compare it with four times the constant term.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Algebraic identities.
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