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 form and bring the skill to full automatism. However, the search for roots does not always make sense, since they may simply not exist. There are special methods for finding roots. In this article, we will analyze the main functions, their scopes, as well as cases where their roots are absent.
Which equation has no roots?
An equation has no roots if there are no such real arguments x for which the equation is identically true. For a non-specialist, this formulation, like most mathematical theorems and formulas, looks very vague and abstract, but this is in theory. In practice, everything becomes extremely simple. For example: the equation 0x=-53 has no solution, since there is no such number x, the product of which with zero would give something other than zero.
Now we will look at the most basic types of equations.
1. Linear equation
An equation is called linear if its right and left parts 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 shown in the illustration below.
Basically, linear equations are solved by simply moving the number part to one part and the contents of x to the other. It turns out an equation of the form mx \u003d n, where m and n are numbers, and x is an 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 there are either infinitely many roots or none at all. With m=0 and n=0, the equation takes the form 0x=0. Absolutely any number will be the solution to such an equation.
But what equation has no roots?
When m=0 and n=0, the equation has no roots from the set of real numbers. 0x=-1; 0x=200 - these equations have no roots.
2. Quadratic equation
A quadratic equation is an equation of the form ax2 + 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 a quadratic equation: D=b2 - 4ac. Then there are two roots x1, 2=(-b ± √D) / 2a.
When D > 0 the equation has two roots, when D=0 - one root. But what quadratic equation has no roots?The easiest way to observe the number of roots of a quadratic equation is on the graph of a function, which is a parabola. At a > 0 the branches are directed upwards, at a < 0 the branches are lowered down. If the discriminant is negative, such a quadratic equation has no roots in the set of real numbers.
You can also visually determine the number of roots without calculating the discriminant. To do this, you need to find the top of the parabola and determine in which direction the branches are directed. You can determine the x-coordinate of a vertex using the formula: x0 =-b / 2a. In this case, the y-coordinate of the vertex is found by simply substituting the x0 value into the original equation.
The quadratic equation x2 – 8x + 72=0 has no roots because it has a negative discriminant D=(–8)2 - 4172=-224. This means that the parabola does not touch the x-axis and the function never takes the value 0, hence the equation has no real roots.
3. Trigonometric equations
Trigonometric functions are considered on a trigonometric circle, but can also be represented in a Cartesian coordinate system. In this article, we will look at two basic trigonometric functions and their equations: sinx and cosx. Since these functions form a trigonometric circle with radius 1, |sinx| and |cosx| cannot be greater than 1. So which sinx equation has no roots? Consider the graph of the sinx function presented in the picturebelow.
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 its modulo is greater than one.
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 used the wrong formula and you need to start all over again. Sometimes, even with the correct finding of the roots, you can forget to take into account the restrictions on the ODZ, 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 with curly or square brackets. Curly braces 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 denote 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 of the system with square brackets is the totality of all the roots of the individual equations. And systems with curly braces have only common roots. Systems of equations can include absolutely diverse functions, so this complexity is notallows you to immediately tell 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 those that do not have them. First of all, if you can't find roots, don't think they don't exist at all. You may have made a mistake somewhere, then just double-check your solution.
We've covered 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 do. To achieve success in solving equations, only attention and concentration are required. Practice more, it 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 a 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 curly brackets if at least one equation has no roots, and with square brackets if all equations have no roots.
