A repeated root (also known as a double root) occurs when a quadratic equation has exactly one unique solution that satisfies the equation. In other words, it means that the quadratic equation touches the x-axis at exactly one point, instead of crossing it at two distinct points. 

For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the presence of a repeated root can be identified by the discriminant (\(\Delta\)):

\[ \Delta = b^2 - 4ac \]

If the discriminant is zero (\(\Delta = 0\)), the quadratic equation has exactly one real root, which is repeated. This means the equation can be factored into a perfect square trinomial:

\[ ax^2 + bx + c = a(x - r)^2 \]

where \( r \) is the repeated root.

### Example:
Consider the quadratic equation:

\[ 9x^2 + 6x + 1 = 0 \]

As we calculated previously, the discriminant for this equation is zero:

\[ \Delta = 6^2 - 4 \cdot 9 \cdot 1 = 36 - 36 = 0 \]

This indicates a repeated root. The root can be found using the quadratic formula:

\[ x = \frac{-b}{2a} = \frac{-6}{2 \cdot 9} = \frac{-6}{18} = -\frac{1}{3} \]

Thus, \( x = -\frac{1}{3} \) is the repeated root. We can rewrite the quadratic equation as:

\[ 9(x + \frac{1}{3})^2 = 0 \]

Or more simply:

\[ (3x + 1)^2 = 0 \]

This shows that the root \( x = -\frac{1}{3} \) is repeated twice, hence the term "repeated root."

Would you like more details or have any questions?

Here are 5 related questions you might ask:
1. What are the implications of having repeated roots in a quadratic equation?
2. How do you factor a quadratic equation with repeated roots?
3. Can you explain the relationship between the discriminant and the nature of the roots?
4. What are some real-world applications of quadratic equations with repeated roots?
5. How do you find the roots of a cubic polynomial?

**Tip:** When solving quadratic equations, always check the discriminant to determine the nature of the roots. A discriminant of zero indicates a repeated root, meaning the parabola touches the x-axis at exactly one point.

Explore the concept of repeated roots in quadratic equations with a detailed explanation. Learn how to identify repeated roots using the discriminant and factorization methods. Suitable for students in Grades 9-12.

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