For the equation \\(x^2-2(k-2)x+(k^2-4)=0\\) to have equal roots, what is the value of \\(k\\)?
Answer and explanation
Correct answer: \\(k=2\\)
A quadratic equation has equal roots when its discriminant satisfies \\(D=b^2-4ac=0\\). Here, \\(a=1, b=-2(k-2), c=k^2-4\\), so \\(D=4(k-2)^2-4(k^2-4)=16(2-k)\\). Setting \\(D=0\\) gives \\(16(2-k)=0\\), hence \\(k=2\\). As a check, when \\(k=2\\), the equation becomes \\(x^2=0\\), which has the repeated root \\(x=0\\). Exam tip: For equal roots, directly apply \\(b^2-4ac=0\\).
Frequently asked questions
What is the correct answer to this question?
\\(k=2\\)
Why is this the correct answer?
A quadratic equation has equal roots when its discriminant satisfies \\(D=b^2-4ac=0\\). Here, \\(a=1, b=-2(k-2), c=k^2-4\\), so \\(D=4(k-2)^2-4(k^2-4)=16(2-k)\\). Setting \\(D=0\\) gives \\(16(2-k)=0\\), hence \\(k=2\\). As a check, when \\(k=2\\), the equation becomes \\(x^2=0\\), which has the repeated root \\(x=0\\). Exam tip: For equal roots, directly apply \\(b^2-4ac=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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