Difference Between Search & Browse Methods in Odoo 17
Β
1.3 solving equations t
1. Solving Equations
Example A. 3x2 β 2x = 8
We solve polynomial equations by factoring.
Set one side to 0, 3x2 β 2x β 8 = 0 factor this
(3x + 4)(x β 2) = 0 extract answers
so x = β4/3, 2
the roots for the equation ax2 + bx + c = 0 are
x =
βb Β± οb2 β 4ac
2a
A βrootβ is a solution for
the equation β# = 0β.
For 2nd degree or quadratic equations, we may also
use the Quadratic Formula (QF):
For 3x2 β 2x β 8 = 0, a = 3, b = β2, c = β8,
so b2 β 4ac = 100, or οb2 β 4ac = 10
Hence x = (2 β 10)/6 or (2 + 10)/6
or x = β4/3, 2
2. Rational Equations
Solve rational equations by clearing all denominators
using the LCD.
Example B. Solve
LCD = (x β 2)(x + 1), multiply the LCD to both sides
of the equation:
]
2(x + 1) = 4(x β 2) + 1*(x + 1)(x β 2)
2x + 2 = 4x β 8 + x2 β x β 2
2x + 2 = x2 + 3x β 10
0 = x2 + x β 12
0 = (x + 4)(x β 3) ο x = -4, 3
Both are good.
x β 2
2 =
x + 1
4
+ 1
(x + 1) (x β 2) (x + 1)(x β 2)
x β 2
2 = x + 1
4 + 1
(x β 2)(x + 1) * [
3. Power Equations
To solve a power equation, take the reciprocal power,
so if xR = c, xp/q = c
x = (Β±)cq/por
Equations of the Form xp/q = c
then x = (Β±)c1/R
Example F. Solve (2x β 3)3/2 = -8
Raise both sides to 2/3 power.
(2x β 3)3/2 = -8 ο [(2x β 3)3/2]2/3 = (-8)2/3
(2x β 3) = 4
2x = 7 ο x = 7/2
Since x = 7/2 doesn't work because 43/2 = -8,
there is no solution.
4. Radical Equations
Solve radical equations by squaring both sides to
remove the square root. Do it again if necessary.
Reminder: (A Β± B)2 = A2 Β± 2AB + B2
Example G. Solve
οx + 4 = ο5x + 4 square both sides;
(οx + 4)2 = (ο5x + 4 )2
x + 2 * 4 οx + 16 = 5x + 4 isolate the radical;
8οx = 4x β 12 divide by 4;
2οx = x β 3 square again;
( 2οx)2 = (x β 3)2
4x = x2 β 6x + 9
0 = x2 β 10x + 9
0 = (x β 9)(x β 1)
x = 9, x = 1 Only 9 is good.
5. The geometric meaning of β|x β y|β or β|y β x|β is
βthe distance between x and yβ.
So the absolute value equation β| x β c | = rβ is claiming
that βthe distance from x to c is rβ. Hence x = c Β± r.
Absolute Value Equations
Example G. Draw and solve for x geometrically
if |x β 7| = 12.
The equation asks for the locations of x's that are
12 units away from 7.
c
r
x = c + r
So to the right x = 7 + 12 = 19.
and to the left x = 7 β 12 = β 5.
7
1212
x = 19x = β 5
r
x = c β r
6. Zeroes and Domains
Example I.
For a reduced fractional formula f = ,
the zeroes of f are where N = 0
the domain of f are where βD = 0β
N
D
x2 β 4.Find the zeros and the domain f = x2 β 1
The zeros of f are where x2 β 1 = 0 or x = Β±1.
The domain of f are
βall numbers x2 β 4 = 0, or all numbers except x = Β±2β.
7. A. Solve the following equations by factoring.
5. x2 β 3x = 10
9. x3 β 2x2 = 0
6. x2 = 4
7. 2x(x β 3) + 4 = 2x β 4
10. 2x2(x β 3) = β4x
8. x(x β 3) + x + 6 = 2x2 + 3x
1. x2 β 3x β 4 = 0 2. x2 β 2x β 15 = 0 3. x2 + 7x + 12 = 0
4. βx2 β 2x + 8 = 0
11. 4x2 = x4
12. 7x2 = β4x3 β 3x 13. 5 = (x + 2)(2x + 1)
14. (x + 1)2 = x2 + (x β 1)2 15. (x + 3)2 β (x + 2)2 = (x + 1)2
B. Solve the following equations by the quadratic formula.
If the answers are not real numbers, just state so.
1. x2 β x + 1 = 0 2. x2 β x β 1 = 0
3. x2 β 3x β 2 = 0 4. x2 β 2x + 3 = 0
5. 2x2 β 3x β 1 = 0 6. 3x2 = 2x + 3
Equations
8. Rational Equations
7. 1
x
+
1
x β 1
=
5
6
8. 1
x
+
1
x + 2
=
3
4
9. 2
x
+ 1
x + 1
= 3
2
10. + 5
x + 2
= 22
x β 1
11. β 1
x + 1
= 3
2
12.6
x + 2
β 4
x + 1
= 11
x β 2
x
6 3
1 2
3
5
2
3β+ = x1. x
4 6
β3 1
8
β5
β 1β = x2.
x
4 5
3 2
10
7
4
3+β = x3. x
8 12
β5 7
16
β5
+ 1+ = x
(x β 20) = x β 3
100
30
100
205.
(x + 5) β 3 = (x β 5)
100
25
100
206.
C. Clear the denominators of the equations, then solve.
9. Radical Equations and Power Equations
D. Isolate one radical if needed, square both sides, do it again
if necessary, to solve for x. Check your answers.
1. οx β 2 = οx β 4 2. οx + 3 = οx + 1
3. 2οx β 1 = οx + 5 4. ο4x + 1 β οx + 2 = 1
5. οx β 2 = οx + 3 β 1 6. ο3x + 4 = ο3 β x β 1
7. 2οx + 5 = οx + 4 8. ο5 β 4x β ο3 β x = 1
E. Solve by raising both sides to an appropriate power.
No calculator.
1. x β2 = 1/4 2. x β1/2 = 1/4
3. x β3 = β8 4. x β1/3 = β8
5. x β2/3 = 4 6. x β3/2 = 8
7. x β2/3 = 1/4 8. x β3/2 = β 1/8
9. x 1.5 = 1/27 10. x 1.25 = 32
11. x β1.5 = 27 12. x β1. 25 = 1/32
10. F. Solve for x.
1. Is it always true that I+x| = x? Give reason for your answer.
2. Is it always true that |βx| = x? Give reason for your answer.
Absolute Value Equations
3. |4 β 5x| = 3 4. |3 + 2x| = 7 5. |β2x + 3| = 5
6. |4 β 5x| = β3 7. |2x + 1| β 1= 5 8. 3|2x + 1| β 1= 5
9. |4 β 5x| = |3 + 2x|
11. |4 β 5x| = |2x + 1| 12. |3x + 1| = |5 β x|
10. |β2x + 3|= |3 β 2x|
Solve geometrically for x. Draw the solution.
13. |x β 2| = 1 14. |3 + x| = 5 15. | β9 + x| = β7
x β 4
G. Find the zeros and the domain of the following rational
formulas. (See 2.1)
2x β 1
1. x2 β 1.
x2 β 43.
5x + 7
2.
3x + 5
x2 β x
x2 β x β 24.
x2 β 4x5.
x2 + x β 2
x2 + 2x6.
x2 + x + 2
2x2 β x β 17.
x3 + 2x
x4 β 4x8.
x3 β 8
16. |2 + x| = 1 17. |3 β x| = β5 18. | β9 β x| = 8