The roots of an equation are the values for the variable(s) that make the equation true. They are also known as solutions. For example, in the equation x + 5 = 10, the root is x = 5 because substituting 5 for x makes the equation true: 5 + 5 = 10. Roots of a Quadratic Equation For a quadratic equation, which has the form ax^2 + bx + c = 0, the roots are the values of x where the parabola (the graph of the equation) crosses the x-axis. At these points, the value of the function is zero. A quadratic equation can have two, one, or no real roots. * Two Real Roots: The parabola intersects the x-axis at two distinct points. * One Real Root: The parabola touches the x-axis at exactly one point (its vertex is on the x-axis). This is also called a repeated root. * No Real Roots: The parabola does not intersect the x-axis at all. In this case, the roots are complex numbers. The roots of a quadratic equation can be found using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} The term under the square root, b^2 - 4ac, is called the discriminant. Its value determines the number and type of roots: * If b^2 - 4ac > 0, there are two distinct real roots. * If b^2 - 4ac = 0, there is one real root. * If b^2 - 4ac < 0, there are two complex roots.

August 20, 2025

Eric