Repeated Roots Calculator

Quadratic equations often have two distinct roots, but sometimes they have repeated roots. This calculator helps you find the repeated roots of quadratic equations in the form ax² + bx + c = 0.

What Are Repeated Roots?

Repeated roots occur when a quadratic equation has exactly one real root with multiplicity two. This happens when the discriminant (b² - 4ac) is zero. The repeated root is given by the formula:

Root = -b / (2a)

This means the quadratic equation touches the x-axis at exactly one point, rather than crossing it at two distinct points. The graph of the quadratic function will have a vertex at this repeated root.

Repeated roots are also called double roots or equal roots. They are different from complex conjugate roots which occur when the discriminant is negative.

How to Find Repeated Roots

To find the repeated roots of a quadratic equation ax² + bx + c = 0:

Identify the coefficients a, b, and c in the equation.
Calculate the discriminant using the formula b² - 4ac.
If the discriminant is zero, the equation has repeated roots.
Use the formula -b / (2a) to find the repeated root.

The repeated root represents the x-coordinate where the parabola touches the x-axis. The y-coordinate at this point is always zero.

Discriminant = b² - 4ac

Repeated Root = -b / (2a)

Example Calculation

Let's find the repeated roots of the equation 2x² - 8x + 8 = 0.

Identify the coefficients: a = 2, b = -8, c = 8.
Calculate the discriminant: (-8)² - 4(2)(8) = 64 - 64 = 0.
Since the discriminant is zero, there is one repeated root.
Calculate the repeated root: -(-8) / (2*2) = 8 / 4 = 2.

The equation 2x² - 8x + 8 = 0 has a repeated root at x = 2.

This means the quadratic equation touches the x-axis at the point (2, 0). The vertex of the parabola is at this point.

FAQ

What is the difference between repeated roots and complex roots?

Repeated roots occur when the discriminant is zero, resulting in one real root with multiplicity two. Complex roots occur when the discriminant is negative, resulting in two complex conjugate roots.

How do I know if a quadratic equation has repeated roots?

A quadratic equation ax² + bx + c = 0 has repeated roots if the discriminant (b² - 4ac) equals zero. If the discriminant is positive, there are two distinct real roots; if negative, there are two complex roots.

What is the significance of repeated roots in real-world applications?

Repeated roots often indicate a situation where a system reaches a stable equilibrium point. In physics, this might represent a point where an object comes to rest after being launched. In economics, it might represent a point where supply and demand are perfectly balanced.

Can a cubic equation have repeated roots?

Yes, a cubic equation can have repeated roots. For example, x³ - 6x² + 12x - 8 = 0 has a repeated root at x = 2. This means the cubic touches the x-axis at x = 2 and crosses it at another point.