Types of Quadratic Equation Roots

Understanding Quadratic Equation Roots

In a quadratic equation $ax^2 + bx + c = 0$ (where $a \neq 0$), the roots of the equation are the values of $x$ that make the equation true. A quadratic equation always has two roots that can be found using the formula:

The value $b^2 - 4ac$ is called the determinant or discriminant (denoted by $D$), which is very important because it determines the type of roots of the quadratic equation.

Different Real Roots

If $D > 0$ (or $b^2 > 4ac$), then the quadratic equation has two different real roots.

Example: $x^2 - 5x + 6 = 0$

• $a = 1, b = -5, c = 6$
• $D = (-5)^2 - 4 \cdot 1 \cdot 6 = 25 - 24 = 1 > 0$
• The roots are: $x_1 = 3$ and $x_2 = 2$

Equal (Repeated) Roots

If $D = 0$ (or $b^2 = 4ac$), then the quadratic equation has one real repeated root (two roots with the same value).

Example: $x^2 - 6x + 9 = 0$

• $a = 1, b = -6, c = 9$
• $D = (-6)^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0$
• The roots are: $x_1 = x_2 = 3$

Imaginary (Non-Real) Roots

If $D < 0$ (or $b^2 < 4ac$), then the quadratic equation has two different complex (imaginary) roots.

Where $i = \sqrt{-1}$ is the imaginary number.

Example: $x^2 + 2x + 5 = 0$

• $a = 1, b = 2, c = 5$
• $D = 2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16 < 0$
• The roots are: $x_1 = -1 + 2i$ and $x_2 = -1 - 2i$

Relationship Between Roots and Coefficients

If $x_1$ and $x_2$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, then:

This is an important relationship that can be used to find the coefficients of a quadratic equation if its roots are known.