If \(x\) is a real number, what is the minimum value of \(p(x)=x^2+14x+53\)?
Answer and explanation
Correct answer: 4
Completing the square gives \(p(x)=x^2+14x+53=(x+7)^2+4\). Since \((x+7)^2\geq 0\), we have \(p(x)\geq 4\), and equality occurs at \(x=-7\). Therefore, the minimum value is 4, so option B is correct. The nearby distractor 2 does not follow from the correct completed-square form. Exam tip: when a quadratic is written as \((x-h)^2+k\) with a positive leading coefficient, its minimum value is \(k\).
Frequently asked questions
What is the correct answer to this question?
4
Why is this the correct answer?
Completing the square gives \(p(x)=x^2+14x+53=(x+7)^2+4\). Since \((x+7)^2\geq 0\), we have \(p(x)\geq 4\), and equality occurs at \(x=-7\). Therefore, the minimum value is 4, so option B is correct. The nearby distractor 2 does not follow from the correct completed-square form. Exam tip: when a quadratic is written as \((x-h)^2+k\) with a positive leading coefficient, its minimum value is \(k\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Polynomials in one variable.
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