The solution of a quadratic equation is its root, which can be found by solving the equation. For the number of roots of quadratic equation, there are three possible situations:
1. Two real roots: If the discriminant of the equation (b? -4ac) is greater than zero, that is, b? -4ac & gt； 0, the equation has two unequal real roots. The root can be solved by the root formula:
x = (-b √(b？ -4ac)/(2a)
Where+stands for two roots, one is positive and the other is negative.
2.a real number root: if the discriminant of the equation is equal to zero, that is, b? -4ac = 0, then the equation has real roots (multiple roots). The solution formula of root is also applicable, but at this time √ (b? -4ac) equals zero, and the result is simplified as:
x = -b / (2a)
3. complex roots of two yokes: if the discriminant of the equation is less than zero, that is, b? -4ac & lt； 0, the equation has no real root, but two complex roots with * * * yoke. At this time, the root can be represented by the representation of complex numbers.
It should be noted that the root of the equation can be a real number or a complex number. To determine the nature of the root, it is necessary to calculate the discriminant of the equation and judge according to the result of the discriminant.
Characteristics of roots of quadratic equation with one variable
1. Number of roots: A quadratic equation can have zero, one or two roots. It depends on the sign of the discriminant (b 2-4ac) of the equation.
★ When the discriminant is greater than zero (b? -4ac & gt； 0), the equation has two unequal real roots.
★ When the discriminant is equal to zero (b? -4ac = 0), the equation has real roots (multiple roots).
★ When the discriminant is less than zero (b? -4ac & lt； 0), the equation has no real root, but has two * * * yoke complex roots.
2. Properties of roots: The roots of quadratic equations can be real numbers or complex numbers. Real roots refer to roots that exist in the range of real numbers, while complex roots refer to complex numbers that contain real and imaginary parts. Discriminant can help determine the type of root.
★ When the discriminant is greater than zero, the roots are two unequal real numbers.
★ When the discriminant is equal to zero, the root is a real number (multiple root).
★ When the discriminant is less than zero, the root is a complex number of two yokes.
3. Relationship of roots: If a quadratic equation with one variable has real roots, then these two roots satisfy a specific relationship.
★ If the two roots of the equation are x 1 and x2 respectively, then x 1+x2 = -b/a and x1* x2 = c/a. ..
These characteristics can help us understand the properties of the roots of quadratic equations in one variable and then apply them to solve practical problems. By analyzing the relationship between discriminant and equation root, we can determine the type of equation solution, and use these characteristics to calculate and deduce.
The roots of a quadratic equation with one variable have many uses in mathematics and practical application. The following are some common application scenarios:
1. Solving geometric problems: The roots of a quadratic equation can be used to solve problems related to geometric shapes, such as calculating the intersection of parabola and coordinate axis and finding the maximum value. By solving the equation, the properties and characteristics of geometric figures can be determined.
2. Physics: In physics, the roots of a quadratic equation can be used to calculate the trajectory of moving objects, the flight time of projectiles, the impact point and other issues. For example, by modeling the equation of motion as a quadratic equation, the position and time of an object can be determined by using the roots of the equation.
3. Engineering and modeling: In the field of engineering and modeling, using the root of a quadratic equation can help solve various problems. For example, in circuit design, we can calculate the parameter values of electronic components or analyze the response of circuits by solving quadratic equations.
4. Economics and finance: In economics and finance, the root of a quadratic equation can be used to analyze economic models, calculate the rate of return and study market behavior. For example, the relationship between cost, profit and price can be determined by solving quadratic equations.
5. Data analysis and fitting: The root of a quadratic equation is also commonly used for data analysis and curve fitting. By fitting the data into a quadratic equation, the best fitting curve can be found, so as to predict, optimize and make decisions.
These are just some common application scenarios. In fact, the roots of quadratic equations with one variable are widely used in various disciplines and fields. Solving the root of the equation can help us understand the essence of the problem, predict the result and make decisions.
Examples of roots of quadratic equations in one variable
When a concrete quadratic equation is given, we can find its root. The following is an example of solving the root of a quadratic equation with one variable:
Example: solving equation x? -5x+6 = the root of 0.
Solution:
1. First, observe the coefficients a, b and c of the equation. In the equation, a = 1, b = -5 and c = 6.
2. then, calculate the discriminant D = b? -4ac. The value of substitution coefficient is d = (-5) 2-4 *1* 6 = 25-24 =1.
3. According to the discriminant value classification discussion:
☆? When D > 0, the equation has two unequal real roots.
☆? When D = 0, the equation has a real number root (multiple root).
☆? When d < 0, the equation has no real root, but has two * * * yoke complex roots.
4. in this example, the discriminant d =1>; 0, so the equation has two unequal real roots.
5. Use the formula X = (-b √ d)/(2a) to find the root of the equation. Substituting the values of coefficients and discriminant, there are:
x 1 =(-(-5)+√ 1)/(2 * 1)=(5+ 1)/2 = 3
x2 =(-(-5)-√ 1)/(2 * 1)=(5- 1)/2 = 2
6. Therefore, the root of the equation x 2-5x+6 = 0 is x 1 = 3, and x2 = 2.
By solving this example, we get two real roots of a quadratic equation. The concrete solution determines the type of root according to the value of discriminant, and calculates the root by formula. In practical problems, a similar solution process can be carried out according to the given equation.