When solving a quadratic equation, what indicates two identical roots?

When solving a quadratic equation, what indicates two identical roots?

• A. A positive discriminant
• B. A negative discriminant
• C. A discriminant equal to zero
• D. None of the above

Answer: (C)

In the context of solving quadratic equations, the discriminant, which is derived from the quadratic formula \( ax^2 + bx + c = 0 \), determines the nature of the roots of the equation. The discriminant is calculated using the formula \( D = b^2 - 4ac \). When the discriminant is equal to zero, it indicates that the quadratic equation has exactly one unique solution, which is known as a double root or two identical roots. This situation occurs because the quadratic graph touches the x-axis at one point rather than crossing it, indicating that the solutions are repeated. In contrast, a positive discriminant signifies two distinct real roots, as it means the quadratic graph intersects the x-axis at two separate points. A negative discriminant indicates that the roots are complex or nonexistent in the real number system, meaning the graph does not touch or cross the x-axis at all. Therefore, understanding the role of the discriminant is crucial for determining the nature of the roots in quadratic equations, with a discriminant equal to zero specifically indicating the presence of two identical roots.

In the context of solving quadratic equations, the discriminant, which is derived from the quadratic formula ( ax^2 + bx + c = 0 ), determines the nature of the roots of the equation. The discriminant is calculated using the formula ( D = b^2 - 4ac ).

When the discriminant is equal to zero, it indicates that the quadratic equation has exactly one unique solution, which is known as a double root or two identical roots. This situation occurs because the quadratic graph touches the x-axis at one point rather than crossing it, indicating that the solutions are repeated.

In contrast, a positive discriminant signifies two distinct real roots, as it means the quadratic graph intersects the x-axis at two separate points. A negative discriminant indicates that the roots are complex or nonexistent in the real number system, meaning the graph does not touch or cross the x-axis at all.

Therefore, understanding the role of the discriminant is crucial for determining the nature of the roots in quadratic equations, with a discriminant equal to zero specifically indicating the presence of two identical roots.