If the discriminant is equal to 0, how many solutions does the quadratic equation have?

If the discriminant is equal to 0, how many solutions does the quadratic equation have?

• A. Two solutions
• B. No solution
• C. One solution
• D. Infinitely many solutions

Answer: (C)

When the discriminant of a quadratic equation is equal to zero, it signifies that the equation has exactly one real solution. The discriminant, expressed as \(D = b^2 - 4ac\), is a crucial part of the quadratic formula, which is used to find the solutions of the quadratic equation in the standard form \(ax^2 + bx + c = 0\). When the discriminant is zero, it indicates that the two solutions of the quadratic formula, given by \(x = \frac{-b \pm \sqrt{D}}{2a}\), converge into a single value because \(\sqrt{D}\) becomes zero. Thus, the solutions simplify to \(x = \frac{-b}{2a}\). This particular case means that the quadratic graph, which is a parabola, touches the x-axis at one point, referred to as a repeated or double root. This situation illustrates that the quadratic equation has a single solution, confirming the choice of one solution as the correct answer.

When the discriminant of a quadratic equation is equal to zero, it signifies that the equation has exactly one real solution. The discriminant, expressed as (D = b^2 - 4ac), is a crucial part of the quadratic formula, which is used to find the solutions of the quadratic equation in the standard form (ax^2 + bx + c = 0).

When the discriminant is zero, it indicates that the two solutions of the quadratic formula, given by (x = \frac{-b \pm \sqrt{D}}{2a}), converge into a single value because (\sqrt{D}) becomes zero. Thus, the solutions simplify to (x = \frac{-b}{2a}). This particular case means that the quadratic graph, which is a parabola, touches the x-axis at one point, referred to as a repeated or double root.

This situation illustrates that the quadratic equation has a single solution, confirming the choice of one solution as the correct answer.