For the quadratic equation \(ax^2+bx+c=0\) with integer coefficients, where \(a\ne0\), which condition ensures that both roots are distinct real irrational numbers? Here, the discriminant is \(\Delta=b^2-4ac\).
Answer and explanation
Correct answer: \(\Delta>0\) and \(\Delta\) is not a perfect square
Using \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\), \(\Delta>0\) gives two distinct real roots. If positive \(\Delta\) is not a perfect square, \(\sqrt{\Delta}\) is irrational, so both roots are irrational. \(\Delta=0\) gives equal roots. Exam tip: check the discriminant first.
Frequently asked questions
What is the correct answer to this question?
\(\Delta>0\) and \(\Delta\) is not a perfect square
Why is this the correct answer?
Using \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\), \(\Delta>0\) gives two distinct real roots. If positive \(\Delta\) is not a perfect square, \(\sqrt{\Delta}\) is irrational, so both roots are irrational. \(\Delta=0\) gives equal roots. Exam tip: check the discriminant first.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Roots of a Quadratic Equation.
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