For the equation \(x^2-2(k+4)x+25=0\) to have equal roots, what are the values of \(k\)?
Answer and explanation
Correct answer: \(k=1\) or \(k=-9\)
A quadratic equation \(ax^2+bx+c=0\) has equal roots when its discriminant \(D=b^2-4ac\) is zero. Here, \(a=1\), \(b=-2(k+4)\), and \(c=25\), so \(D=4(k+4)^2-100=0\). Thus, \((k+4)^2=25\), giving \(k+4=\pm5\). Therefore, \(k=1\) or \(k=-9\), so option A is correct. Exam tip: For equal-root questions, directly apply the condition \(D=0\).
Frequently asked questions
What is the correct answer to this question?
\(k=1\) or \(k=-9\)
Why is this the correct answer?
A quadratic equation \(ax^2+bx+c=0\) has equal roots when its discriminant \(D=b^2-4ac\) is zero. Here, \(a=1\), \(b=-2(k+4)\), and \(c=25\), so \(D=4(k+4)^2-100=0\). Thus, \((k+4)^2=25\), giving \(k+4=\pm5\). Therefore, \(k=1\) or \(k=-9\), so option A is correct. Exam tip: For equal-root questions, directly apply the condition \(D=0\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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