If the parabola of a quadratic equation with rational coefficients cuts the x-axis at two distinct points and its discriminant \(D\) is not a perfect square, what will be the nature of its roots?
Answer and explanation
Correct answer: Two real, irrational and distinct roots
A parabola intersecting the x-axis at two distinct points means \(D>0\), so the roots are real and distinct. For a quadratic equation with rational coefficients, if \(D\) is not a perfect square, then \(\sqrt{D}\) is irrational; hence, from \(x=\frac{-b\pm\sqrt{D}}{2a}\), both roots are irrational. Therefore, option A is correct. Option B would apply when \(D\) is a perfect square. Exam tip: \(D>0\), \(D=0\), and \(D<0\) indicate distinct real, equal real, and non-real roots, respectively.
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What is the correct answer to this question?
Two real, irrational and distinct roots
Why is this the correct answer?
A parabola intersecting the x-axis at two distinct points means \(D>0\), so the roots are real and distinct. For a quadratic equation with rational coefficients, if \(D\) is not a perfect square, then \(\sqrt{D}\) is irrational; hence, from \(x=\frac{-b\pm\sqrt{D}}{2a}\), both roots are irrational. Therefore, option A is correct. Option B would apply when \(D\) is a perfect square. Exam tip: \(D>0\), \(D=0\), and \(D<0\) indicate distinct real, equal real, and non-real roots, respectively.
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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