What is a Double Root?
When \(\Delta = b^2 - 4ac = 0\), the quadratic has exactly one unique solution. This happens when the parabola's vertex lies exactly on the x-axis.
Double Root Calculator
When discriminant equals zero - a single repeated solution
When \(\Delta = 0\), a quadratic has exactly one solution (a double root). The parabola touches but does not cross the x-axis.
Double Root Calculator
Enter coefficients to find double roots or check if your quadratic has a repeated root.
Equation:
\[x^2 - 4x + 4 = 0\]
Results:
Discriminant:\[\Delta = 0\]
Root Type:Double Root (Repeated)
Solution:\[x = 2\]
Factored Form:\[(x - 2)^2 = 0\]
Understanding Double Roots
Graph Interpretation
The parabola touches the x-axis at one point but does not cross it. This is called a "tangent" intersection.
Perfect Square Trinomial
Quadratics with double roots can always be factored as a perfect square: \((x - r)^2 = 0\) where \(r\) is the repeated root.
Worked Examples
\(x^2 - 6x + 9 = 0\)
\(\Delta = 36 - 36 = 0\)
\(x = \frac{6}{2} = 3\)
\((x - 3)^2 = 0\)
\(4x^2 - 4x + 1 = 0\)
\(\Delta = 16 - 16 = 0\)
\(x = \frac{4}{8} = \frac{1}{2}\)
\((2x - 1)^2 = 0\)
\(x^2 + 4x + 4 = 0\)
\(\Delta = 16 - 16 = 0\)
\(x = -2\)
\((x + 2)^2 = 0\)
Root Types Comparison
| Discriminant | Root Type | Graph |
|---|---|---|
| \(\Delta > 0\) | Two distinct real roots | Crosses x-axis twice |
| \(\Delta = 0\) | One double root | Touches x-axis once |
| \(\Delta < 0\) | Two complex roots | Does not touch x-axis |