Good morning class, in last class we talked about Quadratic Equations. Today we are going to study

quadratic equations further. Last class what we have done is, I am going to revise all that for you. We

know what is quadratic equation? Equation which is of the form ax square plus bx plus c is equal to

zero. If its roots are alpha and beta than ax square plus bx plus c can be written as a into x minus

alpha into x minus beta. This concept will be further used in equations of limits and other topics, so

we should know this. Next we know sum of roots is minus b by a, product of root is c by a that is

constant term upon coefficient of x square. Difference of roots is square root of discriminant divided

by modulus of coefficient of x square. Square root of discriminant divided by mod. And if alpha is a

root of ax square plus bx plus c is equal to zero then alpha satisfies the equation. That is a alpha

square plus b alpha plus c is equal to zero. These are the five important concepts on which we solved

many equations.

Let us move further, after that we talked about cubic equations exactly same thing goes on cubic

equations. If alpha, beta, gama are the roots of ax cube plus bx square plus cx plus d then ax cube

plus bx square plus cx plus d can be written as a into x minus alpha into x minus beta into x minus

gama and from here we can find that sum of roots alpha plus beta plus gama is minus b by a, sum of

roots taken two at a time is c by a, and product of roots is minus times d by a. That is minus times

constant term upon coefficient of x cube.

After that we talked about graph of Quadratic polynomial. Graph of Quadratic polynomial functions,

graph of quadratic polynomial looks like this, it is a parabolic graph. It is a parabolic graph, if a is

greater than zero than parabola is upward parabola. Similarly if a is less than zero parabola is

downward parabola and if discriminant is greater than zero then graph cuts x axis at two distinct

points. If discriminant is equal to zero then graph touches x axis at one point and if discriminant is

less than zero than graph does not intersect the x axis. Here we get two distinct roots, here we get

two equal roots and then we get no real root only imaginary roots. Further here what we can notice

is that if discriminant is greater than zero, graph or value of y is positive as well as negative. Because

graph is above x axis and below x axis both of the cases are possible. If discriminant is equal to zero

then graph is always above x axis or it touches x axis. Then we can say that y is always greater than

zero or y is equal to zero. And what is the third case, discriminant less than zero, when discriminant

is less than zero, graph is always above x axis that is function is always positive. We are talking about

the case a greater than zero. Similarly a less than zero also the same.

So, from here what we found was that if a is greater than zero and d is less than zero then ax square

plus bx plus c is greater than zero for all x belonging to real number. That means if a is positive and d

is negative so quadratic value will always be positive. For all x what will be the quadratic value?

Positive. And a is less than zero and d is less than zero then graph will be like this that means always

below x axis. Means value of y and ax square plus bx plus c value, for all x value will be negative. That

is ax square plus bx plus c is less than zero, for all x belong to real numbers or if there is equality in

discriminant then here also equality will be there. If there is no equality in discriminant then here

there will no equality. Done, this is what we have learnt in previous class.

One more concept that we talked at the end of the last class was on basic conditions on roots. When

two roots will be positive and when two roots will be negative. For which x two roots will be

positive, for which constant condition that both the roots will be positive and for which all

conditions will be there where both the roots will be negative. This is the same sheet which we

studied. If both the roots are positive then what will happen? First is d will be greater than or equal

to zero. Roots will be real, roots are positive, positive means real. Now we talk about positive and

real. When are roots real, discriminant greater than zero, what will be the sum, positive and what

will be the product, positive. Roots will be real, sum will be positive and product will also be positive.

All these three conditions simultaneously will hold. Similarly if both the roots are negative then what

will happen, d will be greater than or equal to zero. Roots will be real, sum will be negative and

product will be positive. Exactly similar conditions and then we talked about roots of opposite signs.

Roots will be of opposite sign when product will be negative only one condition is necessary as well

as sufficient. D greater than zero, we need not apply because discriminant becomes positive by

itself. Only one condition is necessary and sufficient, we need not apply other conditions.

To complete the revision, let me solve 2, 3 more questions on same concepts. Solve this, find ‘k’ for

which roots of this equation are of opposite sign. We have to find out value of ‘k’ for which sign of

both the roots should be opposite. One root positive and other root is negative. How many

conditions are there in opposite sign that is one, which we have learnt now. Which one, roots will be

of opposite sign when product.