If \(a>0\), \(c<0\), and \(b\) is any real number, what will be the nature of the roots of the quadratic equation \(ax^2+bx+c=0\)?
Answer and explanation
Correct answer: Two real and distinct roots
Since \(a>0\) and \(c<0\), we have \(ac<0\). Therefore, \(-4ac>0\), and for any real \(b\), the discriminant \(D=b^2-4ac=b^2+(-4ac)>0\). Hence, the equation has two real and distinct roots. Equal roots would require \(D=0\), which is impossible here. Exam tip: When \(ac<0\), the roots are always real and distinct.
Frequently asked questions
What is the correct answer to this question?
Two real and distinct roots
Why is this the correct answer?
Since \(a>0\) and \(c<0\), we have \(ac<0\). Therefore, \(-4ac>0\), and for any real \(b\), the discriminant \(D=b^2-4ac=b^2+(-4ac)>0\). Hence, the equation has two real and distinct roots. Equal roots would require \(D=0\), which is impossible here. Exam tip: When \(ac<0\), the roots are always real and distinct.
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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