If the equation \(x^2+2kx+k^2-4k+8=0\) has real and equal roots, what is the value of \(k\)?
Answer and explanation
Correct answer: \(k=2\)
For a quadratic equation to have real and equal roots, its discriminant must be zero: \(D=b^2-4ac=0\). Here, \(a=1\), \(b=2k\), and \(c=k^2-4k+8\). Thus, \(D=(2k)^2-4(k^2-4k+8)=16(k-2)\). Setting this equal to zero gives \(k=2\), so option A is correct. Exam tip: For equal roots, immediately apply the condition \(D=0\).
Frequently asked questions
What is the correct answer to this question?
\(k=2\)
Why is this the correct answer?
For a quadratic equation to have real and equal roots, its discriminant must be zero: \(D=b^2-4ac=0\). Here, \(a=1\), \(b=2k\), and \(c=k^2-4k+8\). Thus, \(D=(2k)^2-4(k^2-4k+8)=16(k-2)\). Setting this equal to zero gives \(k=2\), so option A is correct. Exam tip: For equal roots, immediately 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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