For the equation \\((r+2)x^2-2(r+5)x+(r+2)=0\\), what is the value of \(r\) for which the roots are real and equal?
Answer and explanation
Correct answer: \(r=-\frac{7}{2}\)
For real and equal roots, the discriminant must satisfy \(D=b^2-4ac=0\). Here, \(a=r+2\), \(b=-2(r+5)\), and \(c=r+2\). Thus, \(D=4(r+5)^2-4(r+2)^2=12(2r+7)\). Setting \(D=0\) gives \(2r+7=0\), so \(r=-\frac{7}{2}\). The value \(r=-2\) is not valid because it makes the coefficient of \(x^2\) zero, so the equation is no longer quadratic. Exam tip: For equal-root questions, set \(D=0\) first and then verify that \(a\neq0\).
Frequently asked questions
What is the correct answer to this question?
\(r=-\frac{7}{2}\)
Why is this the correct answer?
For real and equal roots, the discriminant must satisfy \(D=b^2-4ac=0\). Here, \(a=r+2\), \(b=-2(r+5)\), and \(c=r+2\). Thus, \(D=4(r+5)^2-4(r+2)^2=12(2r+7)\). Setting \(D=0\) gives \(2r+7=0\), so \(r=-\frac{7}{2}\). The value \(r=-2\) is not valid because it makes the coefficient of \(x^2\) zero, so the equation is no longer quadratic. Exam tip: For equal-root questions, set \(D=0\) first and then verify that \(a\neq0\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
