Let \(p\) and \(q\) be real numbers. If \(x^2-2px+(p^2-q^2)=0\) and \(q\neq 0\), what is the nature of the roots of this quadratic equation?
Answer and explanation
Correct answer: Two real and distinct roots
Here, \(a=1\), \(b=-2p\), and \(c=p^2-q^2\). Therefore, the discriminant is \(D=b^2-4ac=4p^2-4(p^2-q^2)=4q^2\). Since \(q\neq 0\), we have \(D=4q^2>0\), so the equation has two real and distinct roots. In fact, the roots are \(p+q\) and \(p-q\). Option B is incorrect because equal roots require \(D=0\). Exam tip: For a quadratic equation, \(D>0\) always indicates two real and distinct roots.
Frequently asked questions
What is the correct answer to this question?
Two real and distinct roots
Why is this the correct answer?
Here, \(a=1\), \(b=-2p\), and \(c=p^2-q^2\). Therefore, the discriminant is \(D=b^2-4ac=4p^2-4(p^2-q^2)=4q^2\). Since \(q\neq 0\), we have \(D=4q^2>0\), so the equation has two real and distinct roots. In fact, the roots are \(p+q\) and \(p-q\). Option B is incorrect because equal roots require \(D=0\). Exam tip: For a quadratic equation, \(D>0\) always indicates two real and distinct 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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