A double root is a solution to a polynomial equation that occurs twice. When you solve an equation and get the same answer repeated, that repeated solution is called a double root (also known as a repeated root). In technical terms, it’s a root with a multiplicity of 2, meaning the factor that produces it appears twice in the polynomial’s factored form.
How Double Roots Work in Algebra
Every polynomial can be broken down into factors. When one of those factors shows up twice, the root it produces is a double root. Take the equation x² − 6x + 9 = 0. This factors into (x − 3)(x − 3) = 0, or (x − 3)² = 0. The only solution is x = 3, but because the factor (x − 3) appears twice, we say 3 is a double root, or a root of multiplicity 2.
Compare that to x² − 5x + 6 = 0, which factors into (x − 2)(x − 3) = 0. Here you get two distinct roots: x = 2 and x = 3, each with multiplicity 1. The difference is that no factor is repeated.
This concept extends beyond quadratics. A higher-degree polynomial like (x − 1)²(x + 4) = 0 has a double root at x = 1 and a single root at x = −4. The exponent on each factor tells you the multiplicity of the corresponding root.
The Discriminant Test for Quadratics
For quadratic equations in the standard form ax² + bx + c = 0, there’s a quick way to check whether a double root exists without factoring. The discriminant, b² − 4ac, tells you the nature of the roots:
- Discriminant greater than zero: two distinct real roots
- Discriminant equal to zero: exactly one real root, which is a double root
- Discriminant less than zero: no real roots (two complex roots)
When the discriminant equals zero, the quadratic formula simplifies to x = −b / (2a). That single value is your double root. For example, in 9x² + 6x + 1 = 0, the discriminant is 6² − 4(9)(1) = 36 − 36 = 0, confirming a double root at x = −6/18 = −1/3.
A double root of a quadratic with real coefficients is always a real number. Complex or imaginary double roots can’t occur in a real-coefficient quadratic because the discriminant would need to be both zero and negative at the same time, which is impossible.
What a Double Root Looks Like on a Graph
The graphical behavior of a double root is distinctive and easy to spot. Instead of crossing the x-axis, the curve touches it and turns back. The graph just barely makes contact with the axis at that point, like a ball bouncing off a floor.
For a quadratic like y = (x − 3)², the parabola dips down to the x-axis at x = 3, touches it, and curves back up. The vertex of the parabola sits right on the x-axis. Contrast this with a single root, where the graph cuts straight through the axis, or a triple root, where the graph flattens out at the axis and then passes through to the other side.
This visual pattern holds for higher-degree polynomials too. Wherever a polynomial has a double root, the graph will kiss the x-axis at that point without crossing it. If you see a curve that touches the x-axis and bounces away, you’re looking at a root of even multiplicity, and multiplicity 2 (a double root) is the most common case.
The Connection to Derivatives
If you’ve encountered calculus, there’s an elegant way to identify double roots. At a double root, both the function and its derivative equal zero at the same point. This makes sense geometrically: the curve is touching the x-axis (so the function is zero) and it’s also at a turning point (so the slope, given by the derivative, is zero too).
For example, with f(x) = (x − 3)² = x² − 6x + 9, the derivative is f'(x) = 2x − 6. At x = 3, both f(3) = 0 and f'(3) = 0. This property gives you a practical test: if you suspect a root might be a double root, plug it into the derivative. If the derivative is also zero there, you’ve confirmed it.
Writing Equations With Double Roots
You may be asked to write a quadratic that has a specific double root. The process is straightforward. If you want a double root at x = r, your factored form is a(x − r)², where “a” is any nonzero constant that controls how wide or narrow the parabola is.
Want a double root at x = −5? Start with (x − (−5))² = (x + 5)² = x² + 10x + 25. If you need a leading coefficient other than 1, multiply through: 3(x + 5)² = 3x² + 30x + 75. Both versions have the same double root at x = −5.
For polynomials of higher degree, you can combine a double root with other roots. A cubic with a double root at x = 2 and a single root at x = −1 would look like a(x − 2)²(x + 1). Expanding gives you the standard form if needed, but the factored version makes the root structure immediately visible.