Irrational equations. Comprehensive guide
Irrational equations are those in which the variable is contained under the sign of the root.
An irrational equation, as a rule, reduces to an equivalent system containing equations and inequalities.
Of the two systems, choose 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 a natural power. When raising an equation to a power, extraneous roots may appear. Therefore, a necessary part of solving the irrational equation is verification.
Tasks and tests on the topic "Irrational equations"
- Irrational equations - Quadratic equations Grade 8
Lessons: 1 Tasks: 9 Tests: 1
- Irrational equations and inequalities - Important topics for the repetition of the exam in mathematics
Tasks: 11
- §4 Application of the properties of functions to the solution of irrational equations
Lessons: 1 Tasks: 13
- §2 Irrational equations - Section 4. Power function class 10
Lessons: 1 Tasks: 9
- Systems of equations - Equations and inequalities grade 11
Lessons: 1 Tasks: 19 Tests: 1
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) the method of raising both sides of the equation to the same degree;
3) the method of introducing new variables.
If you do not monitor the equivalence of transitions, then verification is an essential element of the solution. O.D.Z. in irrational equations will not help you weed out all extraneous roots. Pay attention to it!
When solving irrational equations, as a rule, the following methods are used: 1) transition to an equivalent system (in this case, verification is not necessary); 2) the method of raising both sides of the equation to the same degree; 3) the method of introducing new variables.
Examples.
| x \u003d -1 |
Solution: DLD:
Square both sides of the equation:
x \u003d 6 is included in the ODZ, which means it can be the root of this equation.
Verification:
Solution: DLD
for 2 + 4y - 12 \u003d 0;
y 1 \u003d -6, y 2 \u003d 2.
a) \u003d - 6. There are no solutions, as -6\u003e 0, and 0.
b) \u003d 2,
x - 3 \u003d 4,
x \u003d 7 is included in the DLD.
Irrational equations
Option 1
x
9
5
x
2 2
x \u003d 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 gap to which
zeros of the function f (x) \u003d
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) \u003d 0.
2) (
x3
3
; 2)
; 3)3; 4)
3
2
.
22
x
3
2
10
10
3
xx41
7. Solve the equation
2 \u003d x1. Find 3 ∙ x0 + 2.
2 х
5
\u003d | x + 3 | 2.
1)
2 х
7
2
6. Solve the equation
7. Solve the equation
x 4
4 х
Option 3
6 \u003d 0.
x
17
3 \u003d | x + 2 |.
1. Indicate the gap to which
roots of equation 1+
1) [1; 2]; 2).
2. Indicate the gap to which
zeros of the function f (x) \u003d
2 3x.
3 2 х
\u003d 2x.
x5
2
;
1
2
1) [0.7; 0.7]; 2) (0; 1]; 3) [1; 0); 4) [
1
2
roots of the equation
+ 4 \u003d x.
1) (2; 3); 2) (8; 7); 3) (0; 2); 4) (3; 9).
4. How many roots does the equation have
\u003d 1x².
2 2
x
14
21
11
2
4
x
x
x
1) not a single one; 2) one; 3) two; 4) four.
5. Solve the equation x + 7 \u003d
. Indicate
15 х
true statement about its roots.
55
two roots, and they are of different signs
two roots and they are positive
only one root and he
only one root and he
1)
2)
3)
positive
4)
negative
6. Find the largest root of the equation
Option 4
1. Indicate the gap to which
roots of the equation x +
1) (5; 1); 2) (3; 1]; 3) (2; 1]; 4) (1; 6).
2. Indicate the gap to which
zeros of the function f (x) \u003d
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 gap to which
roots of the equation
1); 2) (1; 3); 3); 4) (2; 0).
4. Indicate the gap to which
roots of the equation
1) (2; 0); 2) (0; 2); 3) (2; 4); 4) (3; 6).
5. Find the smallest root of the equation
\u003d 62x.
\u003d x + 2.
1) (4
)=0.
92
3 х
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 \u003d 2 | x |
64
x
2 \u003d x + 4.
223
x
.
Option 6
Option 5
7
3 х
\u003d x + 3.
1. Indicate the gap to which
roots of the equation
1) (7; 1.5); 2) (2.1; 1]; 3); 4) (2; 8).
2. Indicate the gap to which
zeros of the function f (x) \u003d
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 \u003d 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 gap to which
roots of the equation
1) [6; 5]; 2) [4; 0]; 3); 4).
6. Let x0 be the smallest root of the equation:
\u003d x5.
x5
46
x
x
7. Solve the equation
2 \u003d x + 4. Find 2 ∙ x01.
|4
|49
xx
4x \u003d 3.
1. Indicate the gap to which
zeros of the function f (x) \u003d
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 \u003d 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 gap to which
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) \u003d 0.
+ x \u003d 3.
2 2
4 х
3 х
1
4
3
7
x
x
x2 \u003d 0.
1
5
1)
8
3
; 2)
1
4
; 3)2; 4)
5
4
.
6. Let x0 be the non-positive root of the equation:
24
x
2 \u003d x2. Find 2 ∙ x0 + 1.
x
7. Solve the equation
4 х
13
\u003d | x + 1 | 3.
Job number
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
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