If \(x^2+24x+c=0\) is a perfect-square quadratic, what is the value of \(c\)?
Answer and explanation
Correct answer: 144
A perfect-square trinomial must match \((x+b)^2 = x^2 + 2bx + b^2\). Here the coefficient of \(x\) is 24, so \(2b=24\) giving \(b=12\), and hence \(c=b^2=12^2=144\). Alternatively use discriminant: \(b^2-4ac=24^2-4\cdot1\cdot c=576-4c=0\) which yields \(c=144\). Note: 576 might look tempting because it is \(24^2\), but the constant term must be \(b^2\) with \(2b=24\), not \(24^2\). Exam tip: either complete the square or set the discriminant to zero to solve such questions quickly.
Frequently asked questions
What is the correct answer to this question?
144
Why is this the correct answer?
A perfect-square trinomial must match \((x+b)^2 = x^2 + 2bx + b^2\). Here the coefficient of \(x\) is 24, so \(2b=24\) giving \(b=12\), and hence \(c=b^2=12^2=144\). Alternatively use discriminant: \(b^2-4ac=24^2-4\cdot1\cdot c=576-4c=0\) which yields \(c=144\). Note: 576 might look tempting because it is \(24^2\), but the constant term must be \(b^2\) with \(2b=24\), not \(24^2\). Exam tip: either complete the square or set the discriminant to zero to solve such questions quickly.
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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