If, for real constants p and q, the expression \(x^2+px+q\) can be written as the perfect square of a binomial involving \(x\) and a constant, which relation between \(p\) and \(q\) is necessary?
Answer and explanation
Correct answer: \(q=\frac{p^2}{4}\)
For a perfect square, \(x^2+px+q=(x+\frac p2)^2\). Squaring gives the constant term \(\frac{p^2}{4}\), so \(q=\frac{p^2}{4}\). The choice \(q=p^2\) wrongly ignores halving the coefficient of \(x\). Exam tip: halve the middle coefficient, then square it.
Frequently asked questions
What is the correct answer to this question?
\(q=\frac{p^2}{4}\)
Why is this the correct answer?
For a perfect square, \(x^2+px+q=(x+\frac p2)^2\). Squaring gives the constant term \(\frac{p^2}{4}\), so \(q=\frac{p^2}{4}\). The choice \(q=p^2\) wrongly ignores halving the coefficient of \(x\). Exam tip: halve the middle coefficient, then square it.
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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