What condition between \(p\) and \(q\) is necessary and sufficient for the polynomial \(x^2+px+q\) to be expressible as the square of a binomial with real coefficients?
Answer and explanation
Correct answer: \(p^2=4q\)
Let \(x^2+px+q=(x+a)^2\). Comparing coefficients gives \(p=2a\) and \(q=a^2\), hence \(p^2=(2a)^2=4q\). The condition \(p^2=q\) does not match the middle-term coefficient. Exam tip: square the coefficient of the middle term 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?
Let \(x^2+px+q=(x+a)^2\). Comparing coefficients gives \(p=2a\) and \(q=a^2\), hence \(p^2=(2a)^2=4q\). The condition \(p^2=q\) does not match the middle-term coefficient. Exam tip: square the coefficient of the middle term 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: Factorisation.
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