What exactly is the root of a fractional equation and what is the root of the equation?

denominator; According to the steps of solving the integer equation (move the term, if there are parentheses, the parentheses should be removed, pay attention to the variable sign, merge the similar terms, and the coefficient is 1) to find the value of the unknown; Root check (after finding the value of an unknown number, it is necessary to check the root, because in the process of converting the fractional equation into an integral equation, the range of values of the unknown is expanded, and the root increment may be generated).

If the simplest common denominator is equal to 0, the root is an incremental root. Otherwise this root is the root of the original fractional equation. If the solved root is an additional root, then there is no solution to the original equation.

If the fraction itself is about to be divided, it should also be brought in for inspection.

1 Definition: When the equation is deformed, sometimes a root that is not suitable for the original equation may be generated, and this root is called the additional root of the original equation.

1) Fractional equations.

2) Irrational equations.

Introduction to the 3-fraction equation root increase:

In the process of converting a fractional equation into an integral equation, if the root of the integral equation is such that the simplest common denominator is 0, then this root is called the original fractional equation. x-2

x+2x+2x 2-4x-2 solution:

x-2)^2-16=(x+2)^2

x^2-4x+4-16=x^2+4x+4

x^2-4x-x^2-4x=4+16-4

8x=16x=-2

But x=-2 makes x+2 and x2-4 equal to 0, so x=-2 is the root.

If both sides of the fractional equation are multiplied by the simplest common denominator, the value of the whole common denominator of the fractional equation is not 0, then this solution is the solution of the time-sharing equation, and if the value of the simplest common denominator is 0, then the solution is the root increase.

For example: Set an equation.

a(x)=0

is (x)=0

of the roots, called. x=a

is the addition of the root of the equation; If x=b

is equation b(x)=0

but not a(x)=0

, called x=b

is equation b(x)=0

of lost roots. How to find additional roots.

What roots are when solving fractional equations is often caused by violating the principle of homogeneous solution of equations or being careless when deforming equations.

For example, if you multiply both sides of the equation x 2=0 by x, and change it to x(x 2)=0, the simplest common denominator multiplied by both sides of the equation is 0, and if it is 0, it is the root increase.

Resources.

The root of the equation is all the values of x that satisfy f(x)=0.

The root of the equation f(x) refers to all the values of x that satisfy f(x)=0. The root sum of the unary quadratic equation is different, the root can be a double root, and the solution must be different, if the unary quadratic equation has 2 different roots, it is also called having 2 different solutions.

Quadratic equations can be found using formulas. Cubic and quadratic equations also have formulas for finding roots, but they are more complex and not easy to use. There is no root finding formula for algebraic equations of five or more.

Note: When solving fractional equations, irrational equations, and logarithmic equations, it is necessary to convert them into integral equations, which sometimes produces root additions - the value of unknown numbers that make the original equation meaningless, and the value is not the solution of the original equation.

For multivariate equations, the solution of the equation cannot be said to be the root of the equation. In this case, there is a difference between the solution and the root. Because there is no concept of root in multivariate equations.

The root of an equation is the value of an unknown number that equalizes the left and right sides of the equation. Unary quadratic equations.

Root reconciliation is different in that roots can be heavy roots.

And the solutions must be different, if there are 2 different roots of a quadratic equation, it is also said to have 2 different solutions.

The solution of the equation and the root of the equation are the values of the unknowns that make the left and right sides of the equation equal.

The square root is also called the quadratic root.

For non-negative real numbers.

, refers to a real number equal to a self-multiplication result, expressed as where the square root of a non-negative real number is called the arithmetic square root.

A positive number has two square roots.

0 has only one square root, which is 0 itself; Negative numbers do not have square roots. Example: The square root of 9 is 3 Note: Sometimes we mean the square root of arithmetic.

When an equation is deformed, it is sometimes possible to produce roots that are not suitable for the original equation, which is called the additional root of the original equation.

If the root of a fractional equation is such that the common denominator of the equation is zero, then this root is the additional root of the original equation.

Causes of rooting:

For fractional equations, when the value of the denominator is zero in the fractional equation, it is meaningless, so the fractional equation does not allow the unknowns to take those values that make the value of the denominator zero, that is, the fractional equation itself implies the condition that the denominator is not zero. When a fractional equation is converted into an integral equation, this restriction is removed, in other words, the range of unknowns in the equation is expanded, and if the root of the transformed integral equation happens to be a value other than the allowable value of the unknowns of the original equation, then the root addition will occur.

Both sides of the fractional equation are multiplied by the simplest common denominator fractional equation as an integral equation, and the allowable value of the unknown is expanded, so the solution of the fractional equation is prone to root increases.

For example: Set an equation.

a(x)=0

is made up of equations. b(x)=0

If the roots of the two equations are exactly the same (including the multiple numbers), then the two equations are said to be equivalent. If.

x=a is the equation.

a(x)=0

but not b(x)=0

of the roots, called. x=a is the root of the equation; If x=b

is equation b(x)=0

but not a(x)=0

, called x=b

is equation b(x)=0

of lost roots.

Zenggen, a mathematical noun. It means that in the process of converting fractional equations into integral equations, if the root of the integral equation makes the simplest common denominator 0, (the root makes the integral equation true, and the denominator in the fractional equation is 0), then this root is called the additional root of the original fractional equation.

In the process of converting fractional equations into integral equations, if the root of the integral equation is such that the simplest common denominator is 0, then this root is called the additional root of the original fractional equation.

Good luck.

1. It refers to the value of an unknown number that can make the left and right sides of the equation equal.

2. A rational equation containing an unknown number or an integer containing an unknown number with a width bond in the denominator is called a fractional equation, and the additional root of the fractional equation is not the root of the original fractional equation, but the root of the integral equation formed by removing the denominator from the fractional equation.

3. The solutions of one-dimensional one-dimensional equations, one-dimensional quadratic equations, and fractional equations that can be reduced to the above two kinds of equations are usually called the roots of equations.

The key is to use the cross-multiplication method to transform the form of the fraction difference on both sides of the equal sign into a form without fractions.

Let's see the picture for the specific process.

x 10=x rotten core (x 2+9), turned into x[ (x 2+9)- 10]=0, so x1=0, or (x 2+9)- blind calendar grip 10=0, the latter becomes Moqing x 2=1, x2,3=earth 1

The procedure for finding the root of an equation by the dichotomy is as follows:

function erfenfa(a,b)%a,b is the interval, s=(a+b) 2; ，while b-a>1e-5 if fun(a)*fun(s)>0。 a=s; elseif fun(a)*fun(s)<0

function y=fun(x)

Dichotomy is the method of dividing in two. Let [a,b] be the tight interval of r, and the successive dichotomy is to create the following interval sequence: a0=a, b0=b, and for any natural number n, [an+1, bn+1] or equal to [an, cn], or equal to [cn, bn], the picobalance where cn denotes the midpoint of [an, bn].

In general, for the function f(x), if there is a real number c, when x=c, if f(c)=0, then x=c is called the zero point of the function f(x). Solving the equation requires all zeros of f(x) to be annihilated. First, find that a and b belong to the interval (x,y), so that f(a),f(b) are different signs, indicating that there must be zero points in the interval (a,b), and then find f[(a+b) 2], now suppose f(a)<0, f(b)>0,a<>

If f[(a+b) 2]=0, the point is the zero point, and if f[(a+b) 2]<0, then there is a zero point in the interval ((a+b) 2,b), and (a+b) 2 is assigned to a.

If f[(a+b) 2]>0, then there is a zero point in the interval (a,(a+b) 2), and (a+b) 2 is assigned to b, and the value of the midpoint function continues to be used from .