Choose the correct condition on \(k\) for the equation \(x^2+2(k+1)x+k^2=0\) to have two distinct real roots.
Answer and explanation
Correct answer: \(k>-frac{1}{2}\)
Here, \(a=1\), \(b=2(k+1)\), and \(c=k^2\). Therefore, the discriminant is \(D=b^2-4ac=4(k+1)^2-4k^2=4(2k+1)\). Two distinct real roots require \(D>0\), so \(4(2k+1)>0\), which gives \(k>- rac{1}{2}\). At \(k=-\frac{1}{2}\), \(D=0\), so the roots are equal rather than distinct. Exam tip: use \(D>0\) specifically for two distinct real roots.
Frequently asked questions
What is the correct answer to this question?
\(k>-frac{1}{2}\)
Why is this the correct answer?
Here, \(a=1\), \(b=2(k+1)\), and \(c=k^2\). Therefore, the discriminant is \(D=b^2-4ac=4(k+1)^2-4k^2=4(2k+1)\). Two distinct real roots require \(D>0\), so \(4(2k+1)>0\), which gives \(k>- rac{1}{2}\). At \(k=-\frac{1}{2}\), \(D=0\), so the roots are equal rather than distinct. Exam tip: use \(D>0\) specifically for two distinct 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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