Let \(r\) be a real number. Which statement correctly describes the nature of the roots of the equation \(x^2-(2r+5)x+(r^2+5r+7)=0\)?
Answer and explanation
Correct answer: No real roots
For a quadratic equation, the discriminant is \(D=b^2-4ac\). Here, \(a=1\), \(b=-(2r+5)\), and \(c=r^2+5r+7\). Thus, \(D=(2r+5)^2-4(r^2+5r+7)=-3\). Since \(D<0\) for every real value of \(r\), the equation has no real roots. Therefore, options B and C are impossible, while option D is incorrect because the result is not restricted to \(r=0\). Exam tip: whenever \(D<0\), conclude immediately that the quadratic has no real roots.
Frequently asked questions
What is the correct answer to this question?
No real roots
Why is this the correct answer?
For a quadratic equation, the discriminant is \(D=b^2-4ac\). Here, \(a=1\), \(b=-(2r+5)\), and \(c=r^2+5r+7\). Thus, \(D=(2r+5)^2-4(r^2+5r+7)=-3\). Since \(D<0\) for every real value of \(r\), the equation has no real roots. Therefore, options B and C are impossible, while option D is incorrect because the result is not restricted to \(r=0\). Exam tip: whenever \(D<0\), conclude immediately that the quadratic has no 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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