A quadratic equation is a polynomial equation of the second degree. It has the standard form a x squared plus b x plus c equals zero, where x is the unknown variable we want to solve for, and a, b, and c are coefficients representing known numbers. The key requirement is that coefficient a cannot be zero, otherwise the equation would become linear instead of quadratic.
The quadratic formula provides a direct method to find the roots of any quadratic equation. The formula is x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. The expression under the square root, b squared minus 4ac, is called the discriminant. The discriminant determines the nature of the roots: if it's positive, we get two distinct real roots; if it's zero, we get one repeated real root; and if it's negative, we get two complex roots.
Quadratic functions create parabolic curves when graphed. The general form is y equals a x squared plus b x plus c. Here we see the graph of y equals x squared minus 2x minus 3. The parabola opens upward since the coefficient of x squared is positive. The vertex is at point (1, -4), which is the minimum point of this parabola. The roots or x-intercepts are at (-1, 0) and (3, 0), where the parabola crosses the x-axis.
There are several methods to solve quadratic equations. The first method is factoring, where we express the quadratic as a product of two linear factors. For example, x squared minus 5x plus 6 equals zero can be factored as (x minus 2) times (x minus 3) equals zero, giving us solutions x equals 2 or x equals 3. Another method is completing the square, where we manipulate the equation to create a perfect square trinomial. This method is particularly useful when the quadratic doesn't factor easily.
Quadratic equations have numerous real-world applications. They appear in physics for projectile motion, in engineering for optimization problems, in economics for profit modeling, and in geometry for area calculations. To summarize: quadratic equations have the standard form a x squared plus b x plus c equals zero. The quadratic formula provides a universal solution method. The discriminant determines whether we get real or complex roots. The graphs are always parabolas. Remember that multiple solving methods are available, so choose the most appropriate one for each problem.