 Hello friends welcome again to the session on quadratic equations now in the previous session we understood what is meant by quadratic equations right why the word quadratic is used we learned that in Latin the word quad is coming from the word quadratus which means means square and hence quadratic equations and we also understood that a general form of quadratic equation is ax square plus bx plus c equals zero this is what we learned right this is a quadratic equation and where a is not equal to zero and abc are real numbers are real numbers that is what we learned in this session we are going to understand what is meant by solutions of a quadratic equation what is meant by solution of a quadratic equation so solution is nothing but you can say they are also called roots of roots of quadratic equation okay what are roots of quadratic equation so those values values of variable those values of variable which when plugged in plugged in the equation right equation the equation is satisfied is satisfied that is what is called the solution of a quadratic equation what is meant by plugging in the values and the equation getting satisfied so let us take an example understand it so let us say we have a quadratic equation x square minus 3x plus 2 equals 0 right where here if you see a is equal to 1 clearly b is equal to minus 3 and c is 2 now let us try this value so what I'm saying is let us say putting x is equal to 1 in the equation so if you do that what will you get so x equals to 1 means 1 square minus 3 times 1 plus 2 which is actually equal to 0 that means we can say that x equals to 1 is a solution as a solution slash root whichever word you can use either of them sorry root to the equation given equation okay similarly let us say x is equal to 2 you put x equals to 2 in the same equation you will get 2 square minus 3 times 2 plus 2 which is again equal to 4 minus 6 plus 2 which is equal to 0 so this is also reduced to 0 so hence we say again x equals to 2 is a solution or root correct now let us take another value let us say x equals to 0 put x equals to 0 what will happen it will be 0 square minus 3 times 0 plus 2 which is clearly not equal to 0 so hence we say x equals to 0 is not a solution or root of x square minus 3x plus 2 equals 0 so if you see you try any other number but for x equals to 1 and 2 you will see that it is not getting reduced to 0 so all those values are definitely not roots to the or the solution of the given equation so please understand a quadratic equation quadratic equation can have can have at max at maximum how many roots only two roots it quadratic equation cannot have more than two roots yes it can have a quadratic equation so hence I can write a quadratic equation a quadratic equation can have can have either 0 roots that is no roots no roots or only one root or at max two roots it cannot have more than two roots at any cost right and why is this we can understand this by you know when you see when you when you see through this this particular problem from the angle of polynomials so hence what was a quadratic equation if you see a quadratic equation was nothing but px equals to 0 where px is a quadratic polynomial ax square plus bx plus c right and this was equated to 0 so what at what value of x this entire thing is reduced to 0 can be traced from the curve of x and px right so let us say if you plot x here and this is px here which is equal to y in you know you have studied this so hence if you draw the curve so it will be something like that right quadratic curves are like that so if you see there are two points here and here which is making or with which where the value of the value of px is 0 right so there are two values of x one is here one is here which is at which the value of px is getting reduced to 0 isn't it so hence these two values of x are called solution to the quadratic equation px equals to 0 now any quadratic curve you would have known that any quadratic curve can have three positions one like that so it is not intersecting the x axis at all or it just touches the x-axis or it touches the x-axis twice correct so this is the case where there is no solution no solution so it is not intersecting x-axis so there is no value of x if you see there is no value of x any anywhere in this line there is no value where px is reduced to 0 right there is no value of x where px is 0 why because if px was 0 then it would definitely touch the x-axis so in this case there is one 0 one root or one solution why because it just it is touching exactly at one point only one one value of x exists and this value of x is this which is you know making px 0 and here there are two values of x one here another one here where px is becoming 0 so there are two roots here two roots okay so hence and if you see a quadratic curve you know polynomial has only one change in direction that is so it either you know it has only one change in direction across the number line what does it mean so let us say if the quadratic curve is like that it is increasing it decreasing here and then it will hit its mean minimum value and then it will change the direction like that right so hence it changes direction only one once so hence there are there's a there's a possibility that it cuts the x-axis at max 2 at 2 locations right or it might not cut at all for example here it is not cutting at all right so what is this this is a graph of a graph of px where px is a quadratic polynomial so here it is not cutting at all in another case it could just touch once and in another case it could just cut it at two locations right so there only so at max it can cut at two locations and hence we say there are only two possible values or solutions for a quadratic equation correct so this is what is meant by solution to a quadratic equation let's take a few example and see whether what is meant by let's say solution to a quadratic equation so let us say we have an equation now we are dealing with examples let's say example 1 we have x square minus 7x plus let's say 6 equals to 0 okay so if you put x equals to 1 again so what will you see you'll see 1 minus 7 into 1 plus 6 is again 0 so x equals to 1 is clearly a solution right and what about x equals to 6 if I put x equals to 6 again you'll get 6 square minus 7 times 6 plus 6 which is again equal to 0 so hence the two roots are x equals to 1 and x equals to 6 how I am finding the roots is a different discussion altogether but what is meant by root you now understand let us take another example so let us say I'm saying x square minus 5x plus 4 is equal to 0 okay so if you see let us check x equals to 0 so what is x equals to 0 so if you put x equals to 0 it will become 0 minus 5 into 0 plus 4 which is not equal to 0 so hence x equals to 0 is not the solution solution of what solution of x square minus 5x plus 4 equals to 0 right that means you can you can you can think like this that if you have a polynomial px which is equal to x square minus 5x plus 4 this particular value will not be 0 at x equals to 0 okay so hence clearly the curve will not be 0 at x equals so this is x equals to 0 so hence at this point the curve will be somewhere here okay and and so hence it is something like that yes if you see x equals to 0 if I put x equals to 0 the value is not 0 px value is not 0 not 0 correct so but if we put x equals to 1 again if you see x equals to 1 what will happen it will become 1 square minus 5 times 1 plus 4 this is 0 fantastic so that means this is x equals to 1 at this point px if you see px px is coming out to be 0 at this point right and similarly if you check this point is x equals to 4 at x equals to 4 as well px is 0 right px is coming out to be 0 at x equals to 4 and x equals to 1 okay so this is what is meant by solution to a quadratic equation now in the next few sessions we'll learn what is meant by or sorry how to find out the solution of a given quadratic equation