Nature of roots of a quadratic equation

Symmetric Functions

Solved Examples

A

ax

with

The **±** means you need to do a **plus** and
a **minus** , and so there are normally **two** solutions.

(**b**^{2} - 4ac) is called the **discriminant** ,
because it can "discriminate" between the possible types of answer :
if it is **positive**, you will get **two** solutions

if it is**zero,** you get just **one** solution,

and if it is**negative** you get two solutions that include **imaginary** numbers.

## Nature of roots of a quadratic equation

The nature of the roots of a quadratic equation depends on b^{2} - 4ac:
(1) If b^{2} - 4ac > 0, then the equation has two **real** and **distinct** (or different) roots.

(2) If b^{2} - 4ac = 0, then the equation has two **real** and **equal** roots .

(3) If b^{2} - 4ac < 0, then the equation has no real roots, i.e. the roots are **complex** .

## Sum of the roots of a quadratic equation

In all the above three cases, sum of the two roots = - b/a
## Product of the roots of a quadratic equation

In all the three cases, product of the two roots = c/a

##
Symmetric Functions

Suppose α and β are the roots of a quadratic equation, then x = α and x = β

x – α = 0, and x – β = 0

(x – α) (x – β) = 0

x^{2} – (α + β)x + αβ = 0

x^{2} – (Sum of the roots)x + Product of the roots = 0

x^{2} – Sx + P = 0
Here, **S** = Sum of the roots, and **P** = Product of the roots.
Thus, equation whose roots are **α** and **β** , is **x**^{2} – Sx + P = 0 , which is known as quadratic equation.

(

if it is

and if it is

(2) If b

(3) If b

x – α = 0, and x – β = 0

(x – α) (x – β) = 0

x

x

x

Any expression **f(a,b)** involving two numbers a and b is said to be symmetric if it
remains **unchanged** when a and b are **interchanged** .
[i.e. if f(a,b) = f(b,a)]

Some of the symmetric functions are :

We can express symmetric functions in terms of (α + β) and αβ .

Some examples are:

(1)

(2) α^{3} + β^{3} = (α + β)^{3} - 3αβ(α + β)

(3)

(4)

(5)

Some of the symmetric functions are :

We can express symmetric functions in terms of (α + β) and αβ .

Some examples are:

(1)

(2) α

(3)

(4)

(5)

x = [ -6 ± √(16) ]/10

x = ( -6 ± 4 )/10

x = -0.2 and -1 The solution to the given quadratic equation is

Example

x = [ +6 ± √0 ]/2

x = 3 The solution to the given quadratic equation is

x = [ +3 ± √-7 ]/4 √-7 is an imaginary number, thus there is no real roots for this equation.

Product of the roots = αβ = c/a = 10/1 = 10

α

= (7)

= 343 – 210 = 133

roots are α and 1/α Sum of roots = α + 1/α = -b/a = -6/3 = -2 .......... (1)

Product of roots = α(1/α) = c/a = k/3 ......... (2) From (2), k/3 = α(1/α) = 1

Therefore,