Can the Quadratic Formula Only Provide One Answer? Exploring the Conditions

The quadratic formula, a powerful tool in algebra, always provides two roots based on the discriminant. However, under specific conditions, it can produce one solution. This article delves into these scenarios, exploring the underlying mathematics and graphical representations.

Understanding the Quadratic Equation and Its Roots

A quadratic equation in the form of ax^2 bx c 0 is known to have two roots. These roots represent the points at which the parabola intersects the x-axis. The roots are given by the quadratic formula:

[ x frac{-b pm sqrt{b^2-4ac}}{2a} ]

When the discriminant b^2 - 4ac is positive, there are two distinct real roots. When it is zero, the equation has a repeated root, and when it is negative, the roots are complex and non-real.

Conditions for a Unique Solution

One of the interesting conditions is when the discriminant is zero, denoted by:

[ b^2 - 4ac 0 ]

Under this condition, the expression (sqrt{b^2 - 4ac}) becomes zero, and the roots from the quadratic formula simplify to:

[ x frac{-b}{2a} ]

This results in a single unique real solution, where the parabola touches the x-axis at exactly one point. This scenario is often referred to as a repeated root or a double real root.

Graphical Representation

Graphically, this situation is represented by a parabola that is tangent to the x-axis. At the point of tangency, the vertex of the parabola lies exactly on the x-axis, indicating that the quadratic equation has a root of multiplicity two.

Special Cases and Examples

There are other cases where the quadratic formula might appear to have a unique solution, but there are special constraints:

Linear Equation Form: For example, the equation (x - 1)^2 0 can be factored into ((x - 1)(x - 1) 0), leading to the solution (x 1) which is a repeated root. Factorization Directly to a Single Root: For instance, the equation x^2 - 2x 1 0 can be directly factored as ((x - 1)(x - 1) 0), again leading to the single solution (x 1).

It’s important to note that, for a quadratic equation to have a unique solution, it must be in the form of a repeated root or a factor where the quadratic expression is zero at a single point.

Conclusion

Though the Fundamental Theorem of Algebra dictates that a quadratic polynomial must have two complex roots (counting multiplicity), under certain conditions, the quadratic formula can provide a unique solution. This occurs when the discriminant is zero, indicating a repeated root. Such cases are visually represented by a parabola tangent to the x-axis.

Understanding these conditions is crucial for solving quadratic equations and interpreting their graphical representations accurately.