Let us check step by step:
- A quadratic equation is of the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are numbers called coefficients.
- Rational coefficients means \(a, b, c\) are either integers or fractions (numbers that can be written as \(\dfrac{p}{q}\)).
- Irrational roots means the solutions (values of \(x\)) cannot be written as a simple fraction, like \(\sqrt{2}, \sqrt{3}, \pi\), etc.
- Consider the equation: \(x^2 - 2 = 0\).
- Here, the coefficients are:
- \(a = 1\)
- \(b = 0\)
- \(c = -2\)
All of these are rational numbers.
- Now solve: \(x^2 - 2 = 0 \implies x^2 = 2\).
- Taking square root: \(x = \pm \sqrt{2}\).
- But \(\sqrt{2}\) is an irrational number.
So this quadratic equation has rational coefficients but irrational roots. Hence, the answer is Yes.