A quadratic equation is a second-degree polynomial equation. It has the general form a x squared plus b x plus c equals zero, where a is not equal to zero. The graph of a quadratic equation is a parabola, which can intersect the x-axis at zero, one, or two points.
The quadratic formula is the universal method to solve any quadratic equation. The formula is x equals negative b plus or minus the square root of b squared minus 4 a c, all divided by 2 a. The discriminant, b squared minus 4 a c, determines the nature of solutions. Let's solve an example: 2 x squared minus 4 x minus 6 equals zero.
The discriminant determines the nature of solutions. When D is greater than zero, we have two distinct real solutions, and the parabola crosses the x-axis at two points. When D equals zero, we have one repeated real solution, and the parabola touches the x-axis at exactly one point. When D is less than zero, we have two complex solutions, and the parabola does not intersect the x-axis.
The factoring method works when the quadratic can be written as a product of two binomials. We use the zero product property: if the product of two factors equals zero, then at least one factor must be zero. For example, to factor x squared minus 5 x plus 6, we need two numbers that multiply to 6 and add to negative 5. These numbers are negative 2 and negative 3, giving us the factors x minus 2 and x minus 3.
To summarize, there are three main methods to solve quadratic equations. The quadratic formula works for any quadratic equation and is the most reliable method. Factoring is quick when the equation can be easily factored. Completing the square is useful for understanding the structure of quadratic functions. Always check the discriminant to understand the nature of your solutions. Choose the method that works best for your specific equation!