Learn Real Roots

In this article we are going to learn about real roots .Roots are the value of the variable that satiafies the given equation.It is also called as sollutions of the equations.The solution may be positive,negative or imaginary numbers.Whenthe roots are real values then its called as real roots. An equation which contains more than one terms are squared but no higher power in terms, having the syntax, ax2+bx+c where a represents the numerical coefficient of x2, b represents the numerical coefficient of x, and c represents the constant term

Example: 4x2+7x+18
Identify the Real Roots in Quadratic Equation

A quadratic equation is in the form of ax2+bx+c,

First we need to find the discernment d = b2- 4ac

If d > 0, the roots are real roots and unequal

If d = 0, the roots are real and equal

If d< 0, the roots are imaginary.
Example of Real Roots

Below you can see the example of real roots -

Example1: Solve the quadratic equation

x2+21x+20

Solution:

The equation is in the form of  ax2+bx+c

Where a=1 and b= 21 and c= 20

Find the discernment d = b2- 4ac

d = (21)2 - 4* 1* 20

d = 441 -80

d = 361 > 0

If d > 0, the roots are real roots and unequal

To find the roots, the following steps are used

Step 1: Multiply the coefficient of x2 and the constant term,

1*20 = 20 (product term)

Step 2: Find the factors for the product term

20--- > 1*20 = 20 (factors are 1 and 20)

20 --- > 1 + 20 = 21 (21 is equal to the coefficient of x)

Step 3: Separate the coefficient of x

x2 + 21x+20

x2 + x + 20x + 20

Step 4: The common term x for the first two terms and 20 for the next two terms are taking outside

x(x + 1) + 20 (x + 1)

(x + 1) (x + 20)

Set this value equal to zero

(x +1) = 0; x = -1

(x+20) = 0; x =-20

Roots of the quadratic equations x =-1 and x= -20

Example2: Solve the quadratic equation

x2+4x+4

Solution:

The equation is in the form of  ax2+bx+c

Where a=1 and b= 4 and c= 4

Find the discernment d = b2- 4ac

d = (4)2 - 4* 1* 4

d = 16 -16

d = 0

If d = 0, the roots are real roots and equal

To find the roots, the following steps are used

Step 1: Multiply the coefficient of x2 and the constant term,

1*4 = 4 (product term)

Step 2: Find the factors for the product term

4--- > 2 * 2 = 4 (factors are 2 and 2)

4 --- > 2 + 2 = 4 (4 is equal to the coefficient of x)

Step 3: Separate the coefficient of x

x2 + 4x+4

x2 + 2x + 2x + 4

Step 4: The common term x for the first two terms and 2 for the next two terms are taking outside

x(x + 2) + 2 (x + 2)

(x + 2) (x + 2)

Set this value equal to zero

(x + 2) = 0; x = -2

(x + 2) = 0; x =-2

Roots of the quadratic equations x =-2.

Problem Solving Situations

The problem solving situations is defined as the process by which the new circumstances situations are collected and resolved. It begins with an solving of all the aspects of the situations problems are solved and ends when a satisfactory answer has been found. The problem solving situations are also functioned in every one life .And now let us see about the problem solving situations.



Problems Based on Problem Solving Situations in Algebra

Determine the given Factor: 144x2 – 81 in term using problem solving situations

Solution:

Step1: The given factor is 144x2 – 81



Step2 : The problem solving situation here is to make squares on both sides means we get the values of 144 = 12 and 81 = 9.

Like 144x2 =81x2

144 = 122

81 = 92

Step3: Arrange the terms 144x2 – 81 = (12x)2 – (9)2      this produced by the different of squares formula

Like, (a + b) (a - b) = a2 + b2 where a is denoted by 12x and b is denoted by  9

Step 4: Therefore the factor of  144x2 – 81 are (12x + 9) (12x – 9).

Find:  8x - 1 = 33  in term using problem solving situations

Solution:

We know about the problem solving situation is to add one on both side we get

8x - 1 + 1 = 31 + 1

After simplifying, 8x = 32.

The required solution are x = 4

One more Problem in Problem Solving Situations Using Algebra

In term using problem solving situations

3x - y = -16---------------------- (1)

2x+8y =-28---------------------- (2)

The above problem handles the problem solving situation as rearranging the equation one means we can get the value for  y .

Step 1: The equation 1 can be written as,

3x - y = -16

y = 3x + 16--------------------- (3)

Now we can assume the value of y in equation 2 means and we get answer for x .

Step 2: Assume the y values in equation (2)

2x + 8(3x + 16) = -28

Step 3: Determine the above equation:

2x + 24x + 128 = -28

26x = -156

x = -6

Step 4: Substitute x values in 1st equation

3(-6) - y = -16

-18-y=-16

Y = -2

The required answer are x = -6, y = -2

The problems can be in term using problem solving situations .

Trigonometric ratios of compound Angles

Definition :

The algebraic sum of two or more angles is called a ' compound angle ' .

If A , B , C are angles , then A + B , A - B , A + B - C  , A - B + C , A + B + C , etc., are compound angles .

Formulas

cos(A + B)  =  cosA cosB  -  sinA sinB  for all A , B in  R .
cos(A - B)   =  cosA cosB  + sinA sinB  for all A , B  in  R .
sin(A + B)   =  sinA cosB  +  cosA sinB  for all  A , B in  R.
sin(A - B)    =  sinA cosB   -  cosA sinB  for all A , B  in  R.
tan(A + B)   =  (tanA+tanB)/(1-tanAtanB)   if  none of A , B , A + B is an odd multiple of (pi)/2  .

cot(A + B)   =  (cotBcotA-1)/(cotB+cotA)   if  none of A , B , A + B is an odd multiple of   (pi)/2 .
tan(A - B)    =   (tanA-tanB)/(1+tanAtanB)    if  none of A , B , A - B is an odd multiple of (pi)/2.
cot(A - B)    =   (cotBcotA+1)/(cotB-cotA)    if  none of A , B , A + B is an odd multiple of  (pi)/2.
sin(A + B) sin(A - B)   =  sin2A - sin2B    =    cos2B  -  cos2A .
cos(A + B) cos(A - B)   =  cos2A - sin2B   =   cos2B  -  sin2A


Solved Problems on Trigonometric Ratios of Compound Angles

1) Find the value of sin 105o and cos 165o .

Sol: sin 105o   =   sin (45o + 60o)

=    sin45o  cos60o   +  cos45o sin60o

=   (1/sqrt(2)) (1/2)   +  (1/sqrt(2)) (sqrt(3)/2)

sin105o   =   (sqrt(3)+1)/(2sqrt(2))

Now , cos165o   =   cos (180o - 15o)    =   - cos15o

=   - cos(45o - 30o)

=   - [ cos45o cos30o + sin 45o sin 30o]

=   -[(1/sqrt(2))(sqrt(3)/2)+(1/sqrt(2))(1/2)]

cos 165o   =   -(sqrt(3)+1)/(2sqrt(2))

2) Show that   cos42o + cos78o + cos162o = 0 .

Solution :  cos42o + cos78o + cos162o

=   cos (60o - 18o)  +  cos(60o + 18o)  +  cos(180o - 18o)

=   (cos60o cos18o + sin60o sin18o)  +  (cos60ocos18o - sin60o sin18o)  -  cos18o

=   2 cos60ocos18o  -  cos18o

=   2(1/2) cos18o  -  cos18o  =  0


More Problems on Trigonometric Ratios of Compound Angles:

1)Find the values of sin 15o , cos 15o , tan15o

Sol:

sin15o   =  sin (60o - 45o)   =  sin 60o cos45o  -  cos60o sin45o

=  (sqrt(3)/2)(1/sqrt(2)) -  (1/2)(1/sqrt(2))

sin 15o  =  (sqrt(3)-1)/(2sqrt(2))

cos 15o   =   cos(60o - 45o)   =   cos60o cos45o  +  sin60o sin45o

=   (1/2)(1/(sqrt(2)))   +  (sqrt(3)/2)(1/sqrt(2))

cos15o     =   (sqrt(3)+1)/(2sqrt(2))

tan15o   =   tan(60o - 45o)   =   (tan60^o-tan45^o)/(1+tan60^otan45^o)

=    (sqrt(3)-1)/(2sqrt(2))

tan15o   =   2 -  sqrt(3)

2) Show that  cos100o cos40o  +  sin100o sin40o   =  1/2

Solution :  cos100o cos40o  +  sin100o sin40o

The above equation is in the form of  cosA cosB  +  sinA sinB  which is equal to cos(A - B)

Here A = 100o  and  B = 40o

cos(A - B)  =  cos(100o - 40o)

=  cos(60o)

=  1/2

Hence  cos100o cos40o  +  sin100o sin40o  =  1/2

Solve Two Pairs of Angles Problems

Let us see about solve two pairs of angles problems in this article. Two lines are intersecting to make a two pair of angle. Two lines sharing the common end point is representing as angle. The word angle is derived form the Latin word. In Geometry angle is the one of the figure. In two pair of angle opposite angle are same and parallel.
Two Pairs of Angle:

There are several types of two pairs of angles is available for the geometry. These are

Complementary angle
Supplementary angle
Vertical angle

Example Diagram for Solve Two Pairs of the Angles Problems:

Pair of complementary angle:

The sum of the two angle measurement is equal to 90 is called as complementary angle.

example figure for two pair of the angles

Pair of supplementary angle:

The sum of the two angle measurement is equal to 180 is calling as complementary angle.

example figure for supplementary pair of angles

Pair of vertical angle:

One line crossed by another line is formed by the intersecting is said to be vertical angle.

example figure for vertical pair of angles
Example 1: Solve Two Pairs of Angles Problems:

In the given diagram find the unknown value of the a, b, c.

example for two pair of angle diagram

Solution:

Step 1: angle x is supplement of 80°

Step 2: a + 80° = 180°

a = 180° - 80°

a = 100°

Step 3: In this diagram angle c and 100° are vertical angles.

So, angle c = 100°

Step 4; In this diagram angle b and 80° are vertical angle

So, angle b = 80°

Step 5:

a = 100°

b = 80°

c = 100°


Example 2: Solve Two Pairs of Angles Problems:

In complementary angle, angle A is 37 degree calculate the unknown angle

Solution:

Step 1: Total angle measurement of the complementary angle is 37°

Step 2: Given angle is A = 37°

Step 3: Subtracting given angle measurement from total angle measurement of complementary angle. 90° - 37° = 53°

Step 4: Unknown angle is 53°

Problem Solving Training

In this article we are giving problem solving training and we can understand how to solving problems. It is very helping you to improve your problem solving skills. In mathematical terms we can see different types of problem solving. Here you can learn math terms, Exam practices test, problem solving and online test and our tutor will helps you and you can get free online tutor. Let us see Problem solving training.
Problem Solving:

Let us see few problems and their solving methods.

Problem solving training 1:

Solve:  `12 / 6 + 5 / 3`

Solution:

Above problem is showing fraction addition. So we should solve this problem in fraction addition operation.

Step 1: `12 /6 + 5 / 3`

Here numerator values are same, but denominators are different. So we should take LCM then only we can add both values. (We can take LCM if denominators are different).

Step 2:  `12 / 6 + 5 / 3`

Take LCM 6, 3 (therefore LCM is 6)

we have to change denominators values like 6.

Step 3: `(12*1)/(6*1) = 12/6 , (5*2)/ (3*2) = 10 / 6`

`12/6 + 10 / 6`

Now denominators are same so add both values

Step 4: `12 / 6 + 10 / 6`

`22 / 6`

Therefore `12/ 6 + 5 / 3 = 22 / 6.`

Problem solving training 2:

Solve:    20 ___ 5   = 25

Solution:

Step 1: Given 20 ___ 5   = 25.

Step2: here we find the symbols which are need for this operation.

Step 3: if we put the - (minus) symbols like 20 - 5 = 15 we can get 15.

So minus operation is not accept

Step4: + (plus) is correct operation for this problem.

20 + 5 = 25

Step 5: Therefore + (plus) symbol is making the number sentences true.

Problem solving training 3:

Divide the two fractions `40 -: 1/ 5`

Solution:

Above problem is showing fraction division. So we should solve this problem in fraction division operation.

Step 1: given` 40 -: 1/ 5`

Step 2: It denoted by `40 -: 1/5`

The right hand side denominator will be change like as 5/1 so

=   `40 * 5/1`

= `200`

Step 3: Therefore answer is 200.
Practices Problems:

1) Solve this fraction `40/ 20`      answer: 2

2) Add `2 / 3 + 3/ 2 `                      answer: `13/6`

3) Find missing number 4 , 8  12 , ___ , 20 , 24 , ____ , 32      answer: 16 , 28

Exponential Growth Graph

The exponential growth graph is nothing but the exponential function occurs if the rate of growth is proportional to the functional value and the current value in the functional part. The exponential graph contains some of the equal intervals and can be called as the exponential growth or exponential decay. Now we are going to see about the exponential growth graph.
Problems on Exponential Growth Graph

Determine the exponential growth for the function $3000 and to double at 21/2 % continuously

Solution:

The exponential growth function can be calculated as,

A = Pert

The rate value which can be taken as 0.105

6000 = 3000 e 0.105t

We have to take natural log on both sides we get,

2 = e0.105t

ln 2 = ln e 0.105t

ln 2 = 0.105t (ln e)

ln 2 = 0.105t

By using the calculator the value can be found as,

0.693147 = 0.105t

t = 6.666

Thus it takes 6.66 years to double the money.

Graph:Graph
More Problems on Exponential Growth Graph:

Example 1:

Determine the value of ‘r’ where A = 50 at t = 6 years and P = 10

Solution:

The exponential growth can be calculated as,

A = Pert

50 = 10 e6r

5 = e6r

The logarithm for the above equation given as,

ln 5 = 6r

ln (5)/6 = r

r = 1.6094/6

r = 0.2682

The growth of exponential is 0.2682.

Example 2:

Determine the exponential growth for the function $4000 and to double at 24/2 % continuously

Solution:

The exponential growth function can be calculated as,

A = Pert

The rate value which can be taken as 0.12

8000 = 4000 e 0.12t

We have to take natural log on both sides we get,

2 = e0.12t

ln 2 = ln e 0.12t

ln 2 = 0.12t (ln e)

ln 2 = 0.12t

By using the calculator the value can be found as,

0.693147 = 0.12t

t = 5.77

Thus it takes 5.77 years to double the money.

Graph:

Graph

Relational Algebra Notation

The algebra is a mathematical system that consists of operands and operators. The operand is a variable or value from which new value can be constructed. The operator is a symbol denoted the procedures of that construct new value from given values. The relational algebra is the operand variable that represented relations and operator is designed to relation in the database. In this below details about the relational algebra notation.

Symbolic Notation in Relational Algebra Notation:

The relational algebra is whose operand is variable that represent in the relations and the operator is designed to do the most common things that we need to do with relations in a database. The algebra result can be used to query language for relations. There are following as relational algebra notation.

Symbolic notations are,

SELECT,

PROJECT,

PRODUCT,

JOIN,

UNION,

INTERSECTION,

DIFFERENCE,

RENAME,

Usage of notations in relational algebra notation:

1.      SELECT:

The SELECT notation represent in symbol as σ (sigma).The SELECT operation used to select the particular data details get in the database.

2.      PROJECT:

The PROJECT notation represent in symbol as π (pi). The PROJECT operation used to can provide itself to concision in the particular database.

3.      PRODUCT:

The PRODUCT is representing in symbol as x (times).The PRODUCT is performed combination of tuples in the data base.

4.      JOIN and UNION:

The JOIN is representing in symbol as |x|(bow-tie) and UNION is representing in symbol as U (cub). The JION   used to get particular data in that it has two tuples in restriction of their cartesian product based on the conditions.  The UNION is used to set the rotational attributes are same means at that data provided.

5.      INTERSECTION and DIFFERENCE,

The INTERSECTION and DIFFERENCE have symbols as ∩ and – (minus). The intersection is used to intersection of two relations and provides the common tuples in the relations. The difference used to when applied to relations which first relation has in the tuple but in the second relation.
Example of Relational Algebra Notation:

To find all employees in department cs.

SELECT depname = cs (employee), becomes σdepname = cs (employee).

To find all employees in department cs called jack.

SELECT depname = cs ^ jack = jack(employee), becomes σdepname = cs  ^ jack= jack(employee).