For the equation \((q+3)x^2-2(q-2)x+q=0\), what is the value of \(q\) for which the roots are equal, given that \(q\ne -3\)?
Answer and explanation
Correct answer: \(q=\frac{4}{7}\)
For equal roots, the discriminant must satisfy \(D=b^2-4ac=0\). Here, \(a=q+3\), \(b=-2(q-2)\), and \(c=q\). Therefore, \(D=4(q-2)^2-4q(q+3)=4(4-7q)\). Setting \(D=0\) gives \(4-7q=0\), so \(q=\frac{4}{7}\). The value \(q=-3\) in option D makes the coefficient of \(x^2\) zero, so the equation is no longer quadratic. Exam tip: For equal-root questions, set \(D=0\) and also verify that \(a\ne0\).
Frequently asked questions
What is the correct answer to this question?
\(q=\frac{4}{7}\)
Why is this the correct answer?
For equal roots, the discriminant must satisfy \(D=b^2-4ac=0\). Here, \(a=q+3\), \(b=-2(q-2)\), and \(c=q\). Therefore, \(D=4(q-2)^2-4q(q+3)=4(4-7q)\). Setting \(D=0\) gives \(4-7q=0\), so \(q=\frac{4}{7}\). The value \(q=-3\) in option D makes the coefficient of \(x^2\) zero, so the equation is no longer quadratic. Exam tip: For equal-root questions, set \(D=0\) and also verify that \(a\ne0\).
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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