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Algebra focuses on the rules regarding the operations and relations of constructions and concepts that are arise from them. Algebra covers polynomials, terms, equations, and algebraic structures. Along with geometry, topology, fractions, and number theory, algebra is one of the main branches of pure mathematics. Quadratic equation is a part of algebra and here’s a look at quadratic formula, geometry, quadratic factorization, and applications.
Quadratic Equation Overview
The concept of quadratic equation is a second degree polynomial equation. The general form of a quadratic equation is:
In this equation, x represents a variable whereas a, b, and c are constants with a not equal to 0. If the value of a is 0, the equation will become a linear equation. To elaborate the equation more, a is the quadratic coefficient, b is the linear coefficient, and c is the constant or free term. The term comes from the Latin word quadratus meaning square. The methods for solving these equations include factoring, graphing, Newton’s method, and most commonly, the quadratic formula.
Quadratic formula is a method used to solve a quadratic equation. This formula is the most commonly used method to solve the equation. There are two solutions for a quadratic equation, which have real or complex coefficients. This method is called roots. There is a catch here in that these two solutions sail on two boats. That means they may or may not be real and may or may not be distinct.
Roots of a quadratic formula are given by:
and ± indicate both
These equations give the solutions for the quadratic equation.
In the quadratic equation above, b2-4ac is called the discriminant. The name is given so because this particular entity has the capability to discriminate between the possible types of answers, which include:
If b2-4ac is positive, the result is two real solutions.
If it is a 0, the result is one real solution, with both the answers being same.
If it is negative, the result is a complex solution.
Geometry is the branch of pure mathematics which deals with the questions of shapes, size, relative positions of figures, and properties of space. The word came from two Greek words: geo meaning earth and metri meaning measurement. This particular branch of mathematics is considered to be the oldest mathematical sciences. There is a concept of geometry in algebra which is called Algebraic Geometry. This field is the embodiment of the Cartesian geometry of co-ordinates. If you take the above quadratic equation, the solutions of the same are also the roots of the quadratic function. This can be shown as follows:
As both the quadratic equation and the function depict the values of x, then it can be formulated as:
Likewise in case of the discriminant, if the value is positive, the graph touches the x-axis at two points. If it’s zero, then the touch point will be one, whereas if the result is negative, then the graph will not touch the x-axis.
Let’s now see how factoring is used for quadratic equations. Let’s use the usual quadratic equation i.e. the above equation for this purpose. In this method, the easiest and simplest way to do factorization is to find the common factors. Let’s take a look into a live example.
Let’s say is the factor for the quadratic ax2+bx+c=0 but the condition here is that r should only be the root of the quadratic equation ax2+bx+c=0. From the quadratic formula, it follows that:
If there is a special case i.e. b2-4ac, in which there is only one distinct root for the quadratic, the quadratic polynomial can then be factored as
The quadratic equation can be applied widely. If there is an equation with the variable x having a certain higher degree, the equation can be solved by bringing the equation into the quadratic form. The problem can be easily solved by bringing down the degree of the variable x. The quadratic equation finds its use and applicability in computing the trajectories of projectile motion, electronic amplifier design, architecture, engineering, among many other applications.