Definition of quadratic equation. How to solve it?

A quadratic equation is a type of polynomial equation with degree 2, which means it has the form of ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. The solutions to a quadratic equation are known as the roots of the equation, and they can be found using a method known as "factoring" or "completing the square."

Solving Quadratic Equation:-

One of the most common methods for solving a quadratic equation is factoring, which involves breaking down the equation into two binomials that multiply to give the original equation. For example, the equation x^2 + 5x + 6 = 0 can be factored into (x + 2)(x + 3) = 0. The solutions to the equation are found by setting each binomial equal to zero and solving for x. In this case, the solutions are x = -2 and x = -3.

Another method for solving a quadratic equation is completing the square. This method involves adding and subtracting the square of half of the coefficient of the x term to both sides of the equation and then factoring the resulting expression. For example, the equation 2x^2 + 5x - 3 = 0 can be completed by adding and subtracting (5/2)^2 to both sides, which gives 2x^2 + 5x - 3 + (5/2)^2 - (5/2)^2 = 0. This equation can then be factored into (2x + 5/2)^2 - (5/2)^2 = 0. The solutions to the equation are found by taking the square root of both sides and solving for x. In this case, x = -5/4 ± (5/4)√2.

Quadratic equations have a wide range of applications in the real world. They are often used to model physical phenomena such as the motion of projectiles, the oscillations of spring, and the spread of disease. They are also used in engineering, economics, and other fields to model and solve problems.

One of the key features of quadratic equations is that they have a unique parabolic shape when graphed. This shape is defined by a vertex, or the lowest or highest point on the graph, and the direction of the parabola, which can be either facing up or facing down. The vertex of the parabola is located at the point (-b/2a, f(-b/2a)) where f(x) is the quadratic equation.

Quadratic equations can have one, two, or no solutions. If the discriminant of the equation b^2-4ac is greater than zero, then the equation has two distinct real solutions. If the discriminant equals zero, then the equation has one real solution. If the discriminant is less than zero, then the equation has no real solutions, but two complex solutions.

Quadratic equations have been known and studied for centuries, dating back to ancient Greece and China. The famous Greek mathematician Pythagoras and his followers are known to have studied the properties of quadratic equations, and the Chinese mathematician and inventor, Zu Chongzhi, studied the properties of conic sections, which include parabolas. Today, quadratic equations are still widely studied and used in various fields, and the methods for solving them are taught in high school and college mathematics courses.

In summary, a quadratic equation is a type of polynomial equation with degree 2, which can be solved using factoring or completing the square. The solutions to a quadratic