One Real Solution Calculator

A quadratic equation has one real solution when the discriminant is zero. This calculator helps you find that solution and understand when it occurs.

What is a one real solution?

A quadratic equation in the form ax² + bx + c = 0 has one real solution when the discriminant (b² - 4ac) equals zero. This means the parabola touches the x-axis at exactly one point, called the vertex.

Quadratic equations with one real solution are called "perfect square" equations because they can be factored into (√a x + √c)² = 0.

How to find one real solution

The solution to a quadratic equation with one real solution is found using the quadratic formula:

x = -b / (2a)

This formula gives the x-coordinate of the vertex of the parabola, which is the point where the equation touches the x-axis.

To determine if an equation has one real solution, first calculate the discriminant:

Discriminant = b² - 4ac

If the discriminant equals zero, the equation has one real solution. If it's positive, there are two real solutions. If negative, there are no real solutions.

Example calculation

Consider the equation x² - 6x + 9 = 0:

Identify coefficients: a = 1, b = -6, c = 9
Calculate discriminant: (-6)² - 4(1)(9) = 36 - 36 = 0
Since discriminant = 0, there is one real solution
Apply quadratic formula: x = -(-6) / (2*1) = 6 / 2 = 3

The equation x² - 6x + 9 = 0 has one real solution at x = 3.

Interpretation of results

The one real solution represents the point where the quadratic function touches the x-axis. This occurs when the parabola has its vertex at the x-axis level.

In practical terms, this means there's exactly one point where the quantity described by the quadratic equation equals zero.

Note: When the discriminant is zero, the quadratic equation can be factored into a perfect square, making it easier to solve.

Frequently Asked Questions

What does it mean when a quadratic equation has one real solution?: It means the parabola represented by the equation touches the x-axis at exactly one point, called the vertex. The discriminant is zero in this case.
How do I know if a quadratic equation has one real solution?: Calculate the discriminant (b² - 4ac). If it equals zero, the equation has one real solution.
What's the difference between one real solution and two real solutions?: With one real solution, the parabola touches the x-axis at its vertex. With two real solutions, it crosses the x-axis at two points. With no real solutions, it doesn't cross the x-axis at all.
Can a quadratic equation have one real solution and no other solutions?: Yes, when the discriminant is zero, the equation has exactly one real solution and no other real solutions.
How is the one real solution different from the vertex of the parabola?: The one real solution is the x-coordinate of the vertex when the discriminant is zero. The vertex is the minimum or maximum point of the parabola.