Equations in which the variable is contained under the root sign are called irrational.
An irrational equation, as a rule, is reduced to an equivalent system containing equations and inequalities.
Of the two systems, choose the one that is easier to solve.
If , the equation is equivalent to the equation .
Irrational equations can also be solved by raising both sides of the equation to natural degree. When raising an equation to a power, extraneous roots may appear. Therefore, a necessary part of solving an irrational equation is verification.
Problems and tests on the topic "Irrational equations"
- Irrational equations - Quadratic equations 8th grade
Lessons: 1 Assignments: 9 Tests: 1
- Irrational equations and inequalities - Important Topics For repetition of the Unified State Exam mathematics
Tasks: 11
- §4 Application of properties of functions to solving irrational equations
Lessons: 1 Tasks: 13
- §2 Irrational equations - Section 4. Power function Grade 10
Lessons: 1 Tasks: 9
- Systems of equations - Equations and inequalities 11th grade
Lessons: 1 Assignments: 19 Tests: 1
When solving irrational equations, the following methods are usually used:
1) transition to an equivalent system (in this case, verification is not needed);
2) a method of raising both sides of the equation to the same power;
3) method of introducing new variables.
If you do not monitor the equivalence of transitions, then checking is a mandatory element of the solution. O.D.Z. in irrational equations will not help you weed out all extraneous roots. Pay attention to this!
When solving irrational equations, as a rule, the following methods are used: 1) transition to an equivalent system (in this case, verification is not needed); 2) a method of raising both sides of the equation to the same power; 3) method of introducing new variables.
Examples.
| x = -1 |
Solution: ODZ:
Let's square both sides of the equation:
x = 6 is included in the ODZ, which means it can be the root of this equation.
Examination:
Solution: ODZ
y 2 + 4y - 12 = 0;
y 1 = -6, y 2 = 2.
a)=-6. There are no solutions, because... -6>0, and 0.
b) = 2,
x - 3 = 4,
x = 7 is included in the ODZ.
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Irrational equations
Option 1
X
9
5
x
2 2
x=3.
roots of the equation
1)(∞;1]; 2)(1;5]; 3)(5;10]; 4); 2)[1;2); 3)(2;2]; 4); 3)[2;3]; 4)(2;3].
3. Indicate the interval to which they belong
zeros of the function f(x)=
1)[1;0]; 2)[1;1); 3)[3;1]; 4)[3;1).
4.Find the arithmetic mean of the roots
equations
x45
=0.
x.
2
- x
2
X
1)1; 2)
; 3)2; 4)
6 .
2
5. Find the largest root of the equation
1)=0.
2)(
x3
3
; 2)
; 3)3; 4)
3
2
.
22
X
3
2
10
10
3
xx41
7. Solve the equation
2 =x1. Find 3∙x0+2.
2 х
5
=|x+3|2.
1)
2 x
7
2
6. Solve the equation
7. Solve the equation
x 4
4 х
Option 3
6=0.
X
17
3=|x+2|.
1.Indicate the interval to which they belong
roots of equation 1+
1)[1;2]; 2).
2.Indicate the interval to which they belong
zeros of the function f(x)=
2 3x.
3 2 x
=2x.
x5
2
;
1
2
1)[0.7;0.7]; 2)(0;1]; 3)[1;0); 4)[
1
2
roots of the equation
+4=x.
1)(2;3); 2)(8;7); 3)(0;2); 4)(3;9).
4. How many roots does the equation have?
= 1x².
2 2
x
14
21
11
2
4
X
X
x
1) none; 2) one; 3) two; 4) four.
5.Solve the equation x+7=
. Specify
15 x
a true statement about its roots.
55
there are two roots, and they are of different signs
there are two roots, and they are positive
there is only one root, and it is
there is only one root, and it is
1)
2)
3)
positive
4)
negative
6. Find the largest root of the equation
Option 4
1.Indicate the interval to which they belong
roots of the equation x+
1)(5;1); 2)(3;1]; 3)(2;1]; 4)(1;6).
2.Indicate the interval to which they belong
zeros of the function f(x)=
2 2x.
5
x1
=1.
1
X
1) [
1
2
;
1
2
]; 2) [0.6;0.6]; 3).
X
).
x
52
1
2
3.Indicate the interval to which they belong
roots of the equation
1); 2)(1;3); 3); 4)(2;0).
4. Indicate the interval to which they belong
roots of the equation
1)(2;0); 2)(0;2); 3)(2;4); 4)(3;6).
5. Find the smallest root of the equation
=62x.
=x+2.
1)(4
)=0.
92
3 x
7
5
5
X
X
2 х
7
3
1)
; 2)2; 3)8; 4)
6. Find the sum of the roots of the equation
23
3
.
X
7. Solve the equation 5=2|x|
64
x -
2 =x+4.
223
х
.
Option 6
Option 5
7
3 х
=x+3.
1.Indicate the interval to which they belong
roots of the equation
1)(7;1.5); 2)(2,1;1]; 3); 4)(2;8).
2.Indicate the interval to which they belong
zeros of the function f(x)=
1)[1;0]; 2)(2;1]; 3)(2;0]; 4)(1;+∞).
3. Let x0 be the smallest root of the equation:
x23
x.
2
68
x -
2 =x+6. Find 2x0.
x
1)0; 2)9; 3)4; 4) the equation has no roots.
4. Find the arithmetic mean of the roots
equations
x21
32
x
=0.
7
X
1)1; 2)
5
2
; 3) no roots; 4) 5.
5. Indicate the interval to which they belong
roots of the equation
1)[6;5]; 2)[4;0]; 3); 4).
6. Let x0 be the smallest root of the equation:
=x5.
x5
46
x -
x
7.Solve the equation
2 =x+4. Find 2∙x01.
|4
|49
xx
4x=3.
1. Indicate the interval to which they belong
zeros of the function f(x)=
1)[0.4;0.4]; 2)(0.6;0.6); 3) (0.7;0.7); 4)[
1;0,6].
2.Find the sum of the roots of the equation
2 3x.
x4
64
x -
2 =x+4.
X
1)1; 2)7; 3)6; 4) the equation has no roots.
3. Find the arithmetic mean of the roots
equations
x57
2
1) 7; 2)1; 3)
; 4) no roots.
4. Indicate the interval to which they belong
roots of the equation
1)(6;4); 2)(0;2); 3)(2;5); 4)(4;0).
5. Find the smallest root of the equation
(2
2)=0.
+x=3.
2 2
4 x
3 x
1
4
3
7
X
X
x2 =0.
1
5
1)
8
3
; 2)
1
4
; 3)2; 4)
5
4
.
6. Let x0 be a non-positive root of the equation:
24
x -
2 =x2. Find 2∙x0+1.
x
7. Solve the equation
4 х
13
=|x+1|3.
Job No.
Option 1
Answers "Irrational Equations"
Option 4
Option 2
Option 3
Option 5
Option 6
1
2
3
4
5
6
7
1
1
2
3
1
Ø
2
4
2
3
3
3
16
2
3
2
4
1
1
1
1;15
2
2
4
3
4
1
±19
2
2
3
2
4
3
0
3
1
2
4
1
Ø
9
