Given that \\(\alpha\ne -1\\), what condition on \\(\alpha\\) is necessary for the equation \\((\alpha+1)x^2-2\alpha x+\alpha=0\\) to have real roots?
Answer and explanation
Correct answer: \\(\alpha\le 0\\)
The coefficients are \\(a=\alpha+1\\), \\(b=-2\alpha\\), and \\(c=\alpha\\). Therefore, the discriminant is \\(D=b^2-4ac=4\alpha^2-4\alpha(\alpha+1)=-4\alpha\\). For real roots, \\(D\ge0\\), so \\(-4\alpha\ge0\\), which gives \\(\alpha\le0\\). The value \\(\alpha=0\\) is included, whereas \\(\alpha=-1\\) is already excluded because it would make the quadratic coefficient zero. Exam tip: In parameter-based quadratic questions, apply \\(D\ge0\\) and also verify that the coefficient of \\(x^2\\) is non-zero.
Frequently asked questions
What is the correct answer to this question?
\\(\alpha\le 0\\)
Why is this the correct answer?
The coefficients are \\(a=\alpha+1\\), \\(b=-2\alpha\\), and \\(c=\alpha\\). Therefore, the discriminant is \\(D=b^2-4ac=4\alpha^2-4\alpha(\alpha+1)=-4\alpha\\). For real roots, \\(D\ge0\\), so \\(-4\alpha\ge0\\), which gives \\(\alpha\le0\\). The value \\(\alpha=0\\) is included, whereas \\(\alpha=-1\\) is already excluded because it would make the quadratic coefficient zero. Exam tip: In parameter-based quadratic questions, apply \\(D\ge0\\) and also verify that the coefficient of \\(x^2\\) is non-zero.
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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