Quadratic Equations and Linear Inequalities, Solution of a Quadratic Equation When D Less Than 0, Fundamental Theorem of Algebra, Inequalities (Inequations)

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Solution of a Quadratic Equation when D < 0

Let us consider the following quadratic equation:

(a) Solve for

Here,

The roots are and or

Thus, the roots are complex and conjugate.

(b) Solve for :

or

Here,

The roots are

Here, also roots are complex and conjugate. From the above examples, we can make the following conclusions:

(i) in both the cases

(ii) Roots are complex and conjugate to each other.

Fundamental Theorem of Algebra

  • You may be interested to know as to how many roots does an equation have? In this regard the following theorem known as fundamental theorem of algebra, is stated (without proof). ‘A polynomial equation has at least one root’.

  • As a consequence of this theorem, the following result, which is of immense importance is arrived at.

  • ‘A polynomial equation of degree has exactly roots’

Inequalities (Inequations)

  • Now we will discuss about linear inequalities and their applications from daily life. A statement involving a sign of equality is an equation.

  • Similarly, a statement involving a sign of inequality, or is called an inequalities.

Some examples of inequalities are:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

  • (5) and (7) are inequalities in two variables and all other inequalities are in one variable. (1) to (5) and (7) are linear inequalities and (6) is a quadratic inequalities.

Solutions of Linear Inequalities in One/Two Variables

Solving an inequalities means to find the value (or values) of the variable (s), which when substituted in the inequalities, satisfies it.

In solving inequalities, we follow the rules which are as follows:

1. Equal numbers may be added (or subtracted) from both sides of an inequalities.

Thus (i) if then and

And (ii) if then and

2. Both sides of an inequalities can be multiplied (or divided) by the same positive number.

Thus (i) if and then and

And (ii) if and then and

3. When both sides of an inequalities are multiplied by the same negative number, the sign of inequality gets reversed.

Thus (i) if and then and

And (ii) if and then and

Example 1:

Solve. Show the graph of the solutions on number line.

Solution:

We have

or

Or

Or

Or

Or

The graphical representation of solutions is given in Fig.

The graphical representation of solutions

The Graphical Representation of Solutions

Example 2:

The marks obtained by a student of Class XI in first and second terminal examination are and , respectively. Find the minimum marks he should get in the annual examination to have an average of at least marks.

Solution:

Let be the marks obtained by student in the annual examination. Then

or or

Thus, the student must obtain a minimum of marks to get an average of at least marks.

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