For the equation \(x^2-(r+5)x+5r=0\) to have two real and distinct roots, which of the following conditions is correct?
Answer and explanation
Correct answer: \(r\ne 5\)
Here, \(a=1\), \(b=-(r+5)\), and \(c=5r\). Therefore, the discriminant is \(D=b^2-4ac=(r+5)^2-20r=(r-5)^2\). Two real and distinct roots require \(D>0\). Hence, \((r-5)^2>0\), which gives \(r\ne5\). In option B, \(r=5\) makes \(D=0\), so the roots are equal. Exam tip: For two distinct real roots of a quadratic equation, always check that \(D>0\).
Frequently asked questions
What is the correct answer to this question?
\(r\ne 5\)
Why is this the correct answer?
Here, \(a=1\), \(b=-(r+5)\), and \(c=5r\). Therefore, the discriminant is \(D=b^2-4ac=(r+5)^2-20r=(r-5)^2\). Two real and distinct roots require \(D>0\). Hence, \((r-5)^2>0\), which gives \(r\ne5\). In option B, \(r=5\) makes \(D=0\), so the roots are equal. Exam tip: For two distinct real roots of a quadratic equation, always check that \(D>0\).
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.
