Welcome to Bridge courseWelcome to Bridge course

Expression: Representation of Expression and equation

Expression: Representation of relationship between two (or more)  

i blvariables

Equation : Statement of equality between two expressions

Quadratic expression in xQuadratic expression in x

Quadratic equation in x

are quadratic expressions

are quadratic equations are quadratic equations

Example: quadratic or not

Expanding we get p g g

This is a quadratic equationThis is a quadratic equation

Example: quadratic or not

This is not a quadratic.q

Root: value(s) for which an equation and are the roots ( ) qsatisfies

and are the roots of the equation

When equation satisfies q

When equation does not When equation does not satisfyWhen equation When equation satisfies

Observe the following exampleObserve the following example

Here product of and Here product of and is the quadratic

expressionexpression,

C lConversely,

Factors of

are

Factors of

andare and

Product of two linear Product of two linear expressions is a quadratic expression. Also a quadratic expression has two linear pfactors

Example

Here is a repeated factor pof

Thus

A quadratic expression has A quadratic expression has either two non repeated linear f t h li f t factors or has a linear factor repeated two times.

SiSince

the roots of the equation the roots of the equation

are given by Here the given Here the given quadratic equation has two different rootstwo different roots

Since

the roots of the equation q

are given by

H h d i i Here the quadratic equation has one root. We say that the roots are repeated.

Every quadratic equation has atleast one root and has atleast one root and at most two roots.

Suppose that the roots of a quadratic equation are 1 and 2q q

In general a quadratic equation In general a quadratic equation having m and n as the roots is

E l Fi d th d ti Example: Find the quadratic equation having and as the roots.

Consider

Dividing by we get

Comparing with we getwe get

F t i ti th d

If and are the roots

Factorization method

If and are the roots of

then

C f h b i Converse of the above is also true

E lExample

RootsRoots

Example : Solve the equationExample : Solve the equation

rootsroots

Example (by completing the square)q )

Roots

or

E lExample

rootsroots

General formula

is called the discriminant of denoted by D,

the equation

On discriminant

If is positive then If is positive then is real. In this

h 2 l dcase the 2 roots are real and distinct.

On discriminant

If i th th If is zero then the equation has a unique root

rootroot

On discriminant

If is negative then If is negative then the roots are not real (imaginary) (imaginary).

Example : Solve for

Here Here

R t l d di ti tRoots are real and distinct

Example : Solve for

The roots are given by g y

R t lRoots are equal.

Example : solve for

Imaginary Imaginary roots

Graph of

If then the graph of is of the form is of the form

If then the graph of is of the form is of the form

Graph of a quadratic function Graph of a quadratic function

Roots

Roots x-intercepts p

G h f d i f iGraph of a quadratic function

R t

Root

Root

Root x-intercept

Some points to be remembered

andand

are differentare different

is an identity. True for any x

Example : For what value of the equation the equation

is an identity? is an identity?

Comparing the two equations

Example : Solve for

and

Roots of the equation areRoots of the equation are

I d ti ti ith In a quadratic equation with leading co-efficient 1 , a student reads the co-efficient of x wrongly as 19 , which was actually 16 and , yobtains the roots as -15 and –4 . The correct roots are The correct roots are...

Product of the roots is Since the sum of the roots is 16 the roots are 10 and 6 16 the roots are 10 and 6

THANK YOU                             THE WOODS ARE LOVELY THE WOODS ARE LOVELY

Seetharam udupi

““THE WOODS ARE LOVELY THE WOODS ARE LOVELY DARK AND DEEPDARK AND DEEPBUT I HAVE PROMISES BUT I HAVE PROMISES TO KEEP TO KEEP AND AND MILES TO GO MILES TO GO BEFORE I SLEEP BEFORE I SLEEP BEFORE I SLEEP BEFORE I SLEEP AND AND MILES TO GO MILES TO GO BEFORE I SLEEP”BEFORE I SLEEP” R b   BEFORE I SLEEP”BEFORE I SLEEP”‐ Robert  

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