Can a Quadratic Equation Have Its Discriminant as a Root?

Yes, a quadratic equation can have its discriminant as one of its roots. This intriguing property can be explored by considering the standard form of a quadratic equation:

% math ax^2 bx c 0%

The discriminant D of this quadratic equation is given by:

% math D b^2 - 4ac%

For the quadratic equation to have D as one of its roots, we can express the equation as:

% math ax^2 bx c 0 aD^2 bD c 0%

This scenario requires that there exist values a, b, and c such that the equation holds true. Let's explore this through specific examples.

Example 1: When the Discriminant is Not a Root

Consider the quadratic equation with coefficients a 1, b -5, and c 6:

% math x^2 - 5x 6 0%

The discriminant is:

% math D (-5)^2 - 4 cdot 1 cdot 6 25 - 24 1%

The roots are given by:

% math x frac{-b pm sqrt{D}}{2a} frac{5 pm 1}{2}

This results in the roots:

Here, the discriminant D 1 is not one of the roots.

Example 2: When the Discriminant is Not a Root but Adjusted Coefficients Can Change This

Consider the coefficients a 1, b -2, and c 1:

% math x^2 - 2x 1 0%

The discriminant is:

% math D (-2)^2 - 4 cdot 1 cdot 1 4 - 4 0%

The roots are:

% math x frac{2 pm 0}{2} 1%

Here, the discriminant D 0 is not a root.

Example 3: When the Discriminant Becomes a Root

Consider the coefficients a 1, b -3, and c 2:

% math x^2 - 3x 2 0%

The discriminant is:

% math D (-3)^2 - 4 cdot 1 cdot 2 9 - 8 1%

The roots are:

% math x frac{3 pm 1}{2} 2 text{ and } 1%

Here, the discriminant D 1 is indeed one of the roots.

Conclusion

Thus, it is possible for a quadratic equation to have its discriminant as one of its roots, depending on the specific values of a, b, and c. This is an interesting property that sheds light on the interplay between the coefficients of a quadratic equation and its discriminant.