Axe? + bx + c = 0
Where a, b and c respectively represent coefficients and x represents unknowns.
To calculate the imaginary root of an unary quadratic equation, you can use the following steps:
1. Calculate the value of the discriminant δ, and the formula of δ is: δ = b? - 4ac .
2. Judge the value of discriminant:
A) if δ >; 0, that is, the discriminant is greater than 0, and the equation has two unequal real roots.
B) If δ = 0, that is, the discriminant is equal to 0, then the equation has two equal real roots.
C) if δ
3. If the discriminant δ
A) Let the imaginary root of the equation be x = p+qi, where p and q represent the real part and imaginary part respectively.
B) Substitute the imaginary root into the equation, and simplify the equation into the equation of P and Q by using the complex property of the imaginary root.
C) solving p and q to obtain the concrete form of virtual root.
It should be noted that imaginary roots always appear in pairs and are generally expressed in the form of complex numbers.
The origin of imaginary root of quadratic equation in one variable
The origin of imaginary root of quadratic equation with one variable is related to complex number theory in mathematics.
In the early days, people found that the quadratic equation with one variable may not have a real number solution. For example, the equation X 2+1= 0 cannot find a solution in the real number range. In order to solve this problem, mathematicians introduced the concept of complex number.
Complex numbers consist of real numbers and imaginary numbers, where the imaginary number is defined as i = √(- 1). By introducing the imaginary unit I, we can define the complex number as a+bi, where A and B are the real and imaginary parts respectively.
For a quadratic equation with one variable, if the discriminant δ = b 2-4ac is less than zero, the equation has no real number solution. However, according to the definition of complex number, we can conclude that the equation still has solutions, but these solutions exist in the form of complex numbers. These complex solutions are often called virtual roots.
By introducing the concepts of imaginary unit I and complex number, the solution set of the equation can be described more completely, and the development of mathematics field is also expanded.
To sum up, the origin of imaginary roots of quadratic equation in one variable is closely related to complex number theory, and their introduction makes the equation still have solutions without real number solutions.
Several common applications of imaginary root of quadratic equation in one variable
1. Analytic geometry in the field of mathematics: In analytic geometry, complex roots are an important tool to describe plane figures. Virtual roots can be used to understand the images and properties of quadratic functions more comprehensively, such as the opening direction and focus position of parabola.
2. Signal processing and engineering: In the field of signal processing and engineering, complex domain is often used to describe the frequency response of oscillators, filters and signal transmission. Complex roots provide an effective method for analyzing and designing these systems, especially in frequency domain.
3. Physics: In physics, the imaginary roots of quadratic equations in one variable often appear in the problems of vibration and fluctuation. For example, the displacement of the harmonic oscillator can be described by a quadratic equation, in which the imaginary root represents the * * * vibration frequency and damping characteristics of the system.
4. Statistics and probability theory: In statistics and probability theory, the imaginary root of a quadratic equation is related to normal distribution and complex probability density function. The application of complex roots can help us understand and calculate the characteristics of probability distribution.
5. Control system and circuit analysis: In electronic engineering and control system, the imaginary root of a quadratic equation usually appears in frequency modulation response, filter design, correction and stability analysis. Virtual roots allow us to analyze and design system performance to meet specific requirements.
An example of imaginary root of quadratic equation in one variable
Example: Solve the solution of equation x 2+2x+5 = 0.
Solution: For a given quadratic equation x 2+2x+5 = 0, we can use the root formula to calculate its solution.
First, we need to calculate the discriminant δ = B2-4ac = 2 2-4 (1) (5) = 4-20 =-16.
Because the discriminant δ is less than zero, it shows that the equation has no real number solution, only imaginary number solution.
The imaginary solution can be represented by a complex number with the imaginary unit I, which is denoted as a+bi.
According to the root formula, we can get the solution of the equation as follows:
x =(-b√δ)/(2a)
Substituting the known value, we get:
x = (-2 √(- 16)) / (2* 1)
After simplification, you can get:
x = (-2 4i) / 2
Further simplification, we get:
x = - 1 2i
Therefore, the solutions of the equation x 2+2x+5 = 0 are x =-1+2i and x =-1-2i.
These two solutions are both complex solutions and imaginary solutions of the equation.
Note: in this example, we didn't find the real number solution, but got the imaginary number solution as the solution of the equation. This shows the application of the imaginary root of a quadratic equation.