What is the nature of the roots of the quadratic equation \(x^2-2(k-3)x+k^2-6k+8=0\)?
Answer and explanation
Correct answer: Always real and distinct
Here, \(a=1\), \(b=-2(k-3)\), and \(c=k^2-6k+8\). Therefore, the discriminant is \(D=b^2-4ac=4(k-3)^2-4(k^2-6k+8)=4\). Since \(D=4>0\) for every real value of \(k\), the roots are always real and distinct. Exam tip: \(D>0\) indicates two real and unequal roots, \(D=0\) indicates equal roots, and \(D<0\) indicates non-real roots.
Frequently asked questions
What is the correct answer to this question?
Always real and distinct
Why is this the correct answer?
Here, \(a=1\), \(b=-2(k-3)\), and \(c=k^2-6k+8\). Therefore, the discriminant is \(D=b^2-4ac=4(k-3)^2-4(k^2-6k+8)=4\). Since \(D=4>0\) for every real value of \(k\), the roots are always real and distinct. Exam tip: \(D>0\) indicates two real and unequal roots, \(D=0\) indicates equal roots, and \(D<0\) indicates non-real roots.
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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