Solving equations in mathematics has a special place. This process is preceded by many hours of studying the theory, during which the student learns how to solve equations, determine their type and brings the skill to full automatism. However, far from always the search for roots makes sense, since they may simply not exist. There are special techniques for finding roots. In this article we will analyze the main functions, their areas of definition, as well as cases when their roots are missing.
Which equation has no roots?
An equation has no roots if there are no real arguments x for which the equation is identically true. For a layman, this formulation, like most mathematical theorems and formulas, looks very blurry and abstract, but this is in theory. In practice, everything becomes extremely simple. For example: the equation 0 * x = -53 has no solution, since there is no such number x whose product with zero would give something other than zero.
Now we will consider the most basic types of equations.
1. The linear equation
An equation is called linear if its right and left sides are represented as linear functions: ax + b = cx + d or in a generalized form kx + b = 0. Where a, b, c, d are known numbers, and x is an unknown quantity . Which equation has no roots? Examples of linear equations are presented in the illustration below.
Basically, linear equations are solved by simply transferring the numerical part to one part, and the contents from x to the other. An equation of the form mx = n is obtained, where m and n are numbers, and x is unknown. To find x, it is enough to divide both parts by m. Then x = n / m. Basically, linear equations have only one root, but there are cases when the roots are either infinitely many or not at all. For m = 0 and n = 0, the equation takes the form 0 * x = 0. The solution to such an equation is absolutely any number.
However, which equation has no roots?
For m = 0 and n = 0, the equation has no roots in the set of real numbers. 0 * x = -1; 0 * x = 200 - these equations have no roots.
2. The quadratic equation
A quadratic equation is an equation of the form ax 2 + bx + c = 0 for a = 0. The most common way to solve a quadratic equation is to solve it through the discriminant. The formula for finding the discriminant of the quadratic equation is: D = b 2 - 4 * a * c. Next, there are two roots x 1,2 = (-b Âą âD) / 2 * a.
For D> 0, the equation has two roots; for D = 0, it has one root. But which quadratic equation has no roots? It is easiest to observe the number of roots of the quadratic equation in the graph of the function, which is a parabola. For a> 0, the branches are directed upwards, for a <0, the branches are lowered down. If the discriminant is negative, such a quadratic equation has no roots on the set of real numbers.
You can also visually determine the number of roots without calculating the discriminant. To do this, find the top of the parabola and determine in which direction the branches are directed. The x coordinate of the vertex can be determined by the formula: x 0 = -b / 2a. In this case, the y coordinate of the vertex is found by simply substituting the value x 0 into the original equation.
The quadratic equation x 2 - 8x + 72 = 0 has no roots, since it has a negative discriminant D = (â8) 2 - 4 * 1 * 72 = -224. This means that the parabola does not touch the abscissa axis and the function never takes the value 0, therefore, the equation has no real roots.
3. Trigonometric equations
Trigonometric functions are considered on a trigonometric circle, however, they can also be represented in a Cartesian coordinate system. In this article, we will consider two main trigonometric functions and their equations: sinx and cosx. Since these functions form a trigonometric circle with a radius of 1, | sinx | and | cosx | cannot be greater than 1. So, which sinx equation has no roots? Consider the graph of the sinx function shown in the picture below.
We see that the function is symmetrical and has a repetition period of 2pi. Based on this, we can say that the maximum value of this function can be 1, and the minimum -1. For example, the expression cosx = 5 will not have roots, since it is greater than unity in absolute value.
This is the simplest example of trigonometric equations. In fact, their solution can take many pages, at the end of which you realize that you have used the wrong formula and you need to start all over again. Sometimes, even with the correct location of the roots, you may forget to take into account the constraints on the DLR, because of which an extra root or interval appears in the answer, and the whole answer turns into an erroneous one. Therefore, strictly follow all the restrictions, because not all roots fit into the scope of the task.
4. Systems of equations
A system of equations is a set of equations combined by curly or square brackets. Curly brackets denote the joint execution of all equations. That is, if at least one of the equations has no roots or contradicts the other, the whole system has no solution. Square brackets indicate the word "or". This means that if at least one of the equations of the system has a solution, then the whole system has a solution.
The answer to the system with square brackets is the totality of all the roots of the individual equations. And systems with braces have only common roots. Systems of equations can include completely different functions, so this complexity does not allow us to say right away which equation has no roots.
Generalization and tips for finding the roots of the equation
In problem books and textbooks there are different types of equations: those that have roots and do not have them. First of all, if you cannot find the roots, do not think that they are not at all. Perhaps you made a mistake somewhere, then it is enough to carefully double-check your decision.
We examined the most basic equations and their types. Now you can tell which equation has no roots. In most cases, this is not at all difficult. To succeed in solving equations, only attention and concentration are required. Practice more, this will help you navigate the material much better and faster.
So, the equation has no roots if:
- in the linear equation mx = n, the value m = 0 and n = 0;
- in the quadratic equation, if the discriminant is less than zero;
- in a trigonometric equation of the form cosx = m / sinx = n if | m | > 0, | n | > 0;
- in a system of equations with braces, if at least one equation has no roots, and with square brackets, if all equations have no roots.