 Welcome again friends to another session on quadratic equations and today in this session we are going to discuss another very vital component of this topic and that is we will be discussing upon nature of roots of quadratic equation okay. So the thing is nature of roots or solution. So before we get into that let us take an example and understand one thing. So let me give you an equation. Let us say we have an equation x square plus x plus 1 equals 0. This is a quadratic equation. I have a definition. What are the values of a, b and c here? So a is clearly 1, b is also 1 and c is also 1. So we understood Sridharacharya's rule and quadratic formula in previous sessions. So according to those rules the solution x will be nothing but minus b plus minus under root b square minus 4ac. I am writing this plus minus here because it represents there are two such solutions. In one we will take plus and in other I will take negative right. So by 2a. This is a concise way of writing x is equal to minus b plus root b square minus 4ac upon twice a or x is equal to minus b minus root b square minus 4ac upon 2a. So combinedly we can write this as plus minus okay now. So let us try and solve this equation. So hence here my equation solution will be x equals to minus 1 plus root 1 minus 4 times 1 times 1 right divided by 2 times 1 and this x will be minus 1 minus under root 1 minus 4 times 1 times 1 divided by 2 times 1. I simply deployed a, b and c right values in the given formula. So if you see I will get minus 1 plus under root minus 3 upon 2 and here I will get x is equal to minus 1 minus under root minus 3 upon 2. Now here is a problem what is the problem guys? What is root of minus 3, root of minus 3 we do not know and if you are restricting ourselves to the realm of real numbers there is no value which when multiplied by itself will fetch you minus 3. So hence we hit a problem we are in a problem so how do we even find the roots? So that means this particular thing is coming from the point that anything under the root should not be negative. So hence if it is negative then we will not be getting roots or the real roots of the given quadratic equation. But later on you will study in later grades that these type of numbers are called imaginary numbers. They are unlike real numbers we also have there are a set of numbers called imaginary numbers. These are imaginary numbers but we are dealing with only real numbers right now. So we will say there exist no real roots for this. So hence what is the you know the inference is there exists there exists no real I am writing real solution solution slash roots for equation which equation x square plus x plus 1 equals 0 correct this is what the inference is. So generalization will mean that would there be some more equations where the roots are not real yes there could be many many more so you can you know you know for example you can have equation like 4x square plus 4x plus 2 equals to 0 this is also you know this will have no real solution another example could be x square plus 6x plus 12 is equal to 0. So you also have no real solution you can check you will always get a negative number under the root so hence no real solution. So hence let us now analyze it from a general perspective so what do I mean so let us say our roots of a equation by Sridharacharya's rule was x is equal to minus b plus under root b square minus 4ac b square minus 4ac upon twice a this was the first root and the second root was x equals to minus b minus root of b square minus 4ac upon twice a. Now clearly since we are dealing with a quadratic equation ax square plus bx plus c equals to 0 we have and was given that a is not equal to 0 so we are safe in terms of so hence clearly if the denominator here becomes 0 then also it's a problem right but that is dealt away with the fact that a is not equal to 0 so for this x to be real we must have this quantity b square minus 4ac to be greater than 0 ok or even if it is 0 it will be you know fine so hence we get three cases case number one one so I am writing case one what is this case this case one means when d what is d d is equal to b square minus 4ac is greater than 0 guys so if it is greater than 0 then we say that we have real solution real roots possible right two real roots you will get two real roots and what are they this they are x is equal to minus b plus root over d upon 2a and x will be minus b minus root over d upon 2a so since d is greater than 0 you will get two such roots this one and this one right case two could be case two could be when d is equal to 0 this also possible when d is 0 so that that then what what does it mean then the roots x will be nothing but minus b plus root d which is 0 upon 2a first root and second root let's say I am calling it as x1 and x2 so second root is x2 minus b minus root 0 upon 2a what do you see you see there here is x1 is equal to x2 right both roots are equal so hence we get we say real we have real roots because both x1 and x2 are real minus b by 2a so we say real and equal roots equal roots or solution right so we say both roots are real and equal and then third case is case 3 case 3 is when d is less than 0 then clearly we can't so root d is not defined root d is then not defined within the set of within the set of real numbers isn't it so we declare what in this case the inference is we have two non real roots non real or you can also say imaginary imaginary or rather the better word is complex roots complex roots okay so two non real roots right so their roots are not real or roots are not real not real right this is the inference two non real roots you when you know the theory or the concept of complex numbers then you will be able to appreciate that even in this case there exist two roots but those two roots definitely are not real but they are called complex roots which an imaginary root is one subset of complex roots so you will see the complex number has two form two parts one imaginary part another real part so hence you can say you know the two real the roots are not real but they are complex this you will study in later grades now how does you know we also want to visualize this you know somehow so what does this mean so if you remember our equation ax square plus dx plus c is equal to 0 this is our equation can be said to be px is equal to 0 where px is a quadratic polynomial right quadratic polynomial ax square plus dx plus c okay so this is the this is the thing so quadratic equation is nothing but you equate a quadratic polynomial to 0 you will get a quadratic expression equation now we know that every quadratic or any polynomial for that matter can be expressed or can be visualized in a graph paper okay so if you really plot a graph of this okay this equation let us say and I am assuming that a is greater than 0 why am I assuming this because it has some relevance in the nature of the graph itself so a greater than 0 the graph opens you know towards positive y axis and if a is a negative less than 0 then it opens towards negative y axis that is a separate discussion but right now let us assume that a is greater than 0 so hence it can have three such visualizations so this is one case right the graph is like this so you know if you have not studied how to express polynomial graphs then what you can do is you can go through our session on polynomials there I have discussed in detail how to express a quadratic polynomial on a graph paper so there are three possibilities one is this another possibility is that the graph touches the x axis exactly at one point and the third possibility is the graph intersects the two intersects the x axis at two points so these are the two you know this is x and this is y axis and now this graph is or is of px okay y is equal to px if I if I plot this you will get three such cases now this case this is case three guys so if you see there is no point where the y value is intersecting x axis meaning what there is no x where y is becoming zero y is always above zero right so the minimum possible value is this and here also x sorry y value is more than zero right so there is no x so I can write in this is case three so in case three in case three there is no x where y is zero hence we say that since it doesn't intersect on the x axis and we say the roots do not exist right so roots were nothing but where this polynomial graph intersects the x axis why why are we calling that as root because if you see this is nothing but y is equal to zero line x axis is called y is equal to zero line and we are equating y is equal to px to zero isn't it so hence wherever wherever the curve intersects y is equal to zero that is x line we will get a root okay but in case three it's not happening in case two if you see there is exactly one point where the polynomial curve is intersecting x axis so hence in case two we say real and equal or coincident root both roots are coinciding one on one top of the other real and real and coincident coincident right or real and equal equal whichever way you can say and in the third case guys if you see we have two distinct two distinct roots at two locations that means there are two values of x so I can summarize case two and we can say um oh sorry case one this is case one so this is case one this is case one so I can say there are two distinct distinct real roots why because you are getting two values of xc one is here and another one is here so two values of x are there where y is becoming zero or px is becoming zero hence these are the three conditions right there is absolutely no other condition so either so hence you can summarize somebody you can write a quadratic equation a quadratic equation equation can have can have two distinct two distinct or two equal or two non real non real roots depending upon depending upon depending upon whether whether d is greater than zero d is equal to zero or d is less than zero right so hence what are the nature of the roots then so if d is less greater than zero nature of root is two real and distinct two real plus distinct distinct means there are two different values if d is equal to zero nature is two equal real roots equal real roots right roots are there but they're equal and real and then there are two non real non real complex roots complex right so in any every case you will get two roots but depending upon which case depending upon d is greater than zero you'll get two real and distinct d less than zero d is equal to zero you'll get two real and equal so you can say this is only one root on low on also but in technical term we say there are two real roots both are coincident and d less than zero are two non real complex roots this is what is all about nature of roots right so in the subsequent you know chess ends we will take up questions related to that