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 a cannot be zero. The graph of a quadratic equation forms a parabola, which is a U-shaped curve.
There are different types of quadratic equations. A complete quadratic has all three terms: a x squared, b x, and c. Incomplete quadratics are missing one term. Pure quadratics have no linear term, while others may have no constant term. Each type creates a different parabola shape.
There are several methods to solve quadratic equations. Factoring works when the equation can be written as a product of two binomials. The quadratic formula works for any quadratic equation. The discriminant tells us how many real solutions exist. When graphed, the solutions are where the parabola crosses the x-axis.
Parabolas have several important properties. The vertex is the highest or lowest point. The axis of symmetry is a vertical line through the vertex. The coefficient a determines if the parabola opens up or down, and how wide or narrow it is. The y-intercept occurs where the parabola crosses the y-axis.
To summarize what we have learned about quadratic equations: They are second-degree polynomial equations that can be solved using multiple methods. Their graphs form parabolas with distinct properties. Quadratic equations are fundamental in mathematics and have numerous practical applications in science and engineering.