For the equation \(3x^2-2(2k+1)x+(k+1)^2=0\) to have two distinct real roots, which condition must \(k\) satisfy?
Answer and explanation
Correct answer: \(k<1-\sqrt{3}\) या \(k>1+\sqrt{3}\)
A quadratic equation has two distinct real roots only when its discriminant satisfies \(D>0\). Here, \(a=3\), \(b=-2(2k+1)\), and \(c=(k+1)^2\). Thus, \(D=b^2-4ac=4(2k+1)^2-12(k+1)^2=4(k^2-2k-2)\). Therefore, \(k^2-2k-2>0\), or equivalently \((k-1)^2>3\), which gives \(k<1-\sqrt{3}\) or \(k>1+\sqrt{3}\). At the endpoint values, \(D=0\), so the roots are equal. Exam tip: for two distinct real roots, first impose the condition \(D>0\).
Frequently asked questions
What is the correct answer to this question?
\(k<1-\sqrt{3}\) या \(k>1+\sqrt{3}\)
Why is this the correct answer?
A quadratic equation has two distinct real roots only when its discriminant satisfies \(D>0\). Here, \(a=3\), \(b=-2(2k+1)\), and \(c=(k+1)^2\). Thus, \(D=b^2-4ac=4(2k+1)^2-12(k+1)^2=4(k^2-2k-2)\). Therefore, \(k^2-2k-2>0\), or equivalently \((k-1)^2>3\), which gives \(k<1-\sqrt{3}\) or \(k>1+\sqrt{3}\). At the endpoint values, \(D=0\), so the roots are equal. Exam tip: for two distinct real roots, first impose 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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