Question

If a,b and c are integers and b^2-4ac is the square root of an integer, explain why the equation has rational roots?

Answer 1

The equation has rational roots because if
`\(b^2-4ac\)` is the square root of an integer, the discriminant
`\(\Delta = b^2-4ac\)` is a perfect square, ensuring that the roots of the quadratic equation are rational.

Step-by-step explanation:

When solving a quadratic equation of the form
`\(ax^2 + bx + c = 0\)`, the discriminant
`\(\Delta = b^2-4ac\)` plays a crucial role. The discriminant determines the nature of the roots:

- If
`\(\Delta > 0\)`, the roots are real and distinct.

- If
`\(\Delta > 0\)`,, the roots are real and equal.

- If
`\(\Delta > 0\)`,, the roots are complex conjugates.

Now, when
`\(b^2-4ac\)` is the square root of an integer, it means that the discriminant is a perfect square. In this case, the square root is rational, leading to rational roots for the quadratic equation. Rational roots occur when the quadratic formula involves the square root of a perfect square, ensuring that the roots are fractions and, therefore, rational.