Solving quadratics by completing the square: no solution

use completing the square to find the roots of the quadratic equation right here now when anyone talks about roots this just means find find the X's where Y is equal to zero that's what a root is a root is an x-value that will make this quadratic function equal zero that will make y equal zero so to find the x's let's just make y equal zero and then solve for x so we get zero is equal to four x squared plus forty x plus 280 now a first step that we might want to do just because it looks like all four of these are divisible or all three of these terms are divisible by four it's just divide both sides of this equation by four that'll make our math a little bit simpler so let's just divide everything by 4 here if we just divide everything by 4 we get 0 is equal to x squared plus 10 X this is 10 X plus 10 X plus 280 divided by 4 is 70 plus 70 now they say confused completing the square actually let me write that 70 a little bit further out and you'll see why I did that in a second so let me just write a plus 70 over here just have kind of an awkward space here you'll see what I'm about to do with this space that has everything to do with completing the square so they say use completing the square which means turn this if you can into a perfect square turn at least part of this expression into a perfect square and then we can use that to actually solve for X so how do we turn this into a perfect square well we have a 10 X here and we know that we can turn this into a perfect square trinomial if we take half of the 10 which is 5 and then we square that so half of 10 is 5 you square it you add a 25 now you can't just willy-nilly add a 25 to one side of the equation without doing something to the other or without just subtracting the 25 right here right think about it I have not changed the equation I have added 25 and I subtracted 25 so I've added nothing to the right-hand side I could add a billion and subtract a billion and not change the equation so I have not change the equation at all right here but what I have done is I've made it possible to express these these three terms as a perfect square that right there two times five is ten five squared is 25 so that is X plus five X plus five squared and if you don't believe me multiply it out you're going to have a x squared plus five x plus five X which will give you ten X plus 5 squared which is 25 so those first three terms become that and then the second three terms I'm sorry the second two terms right there you just add them let's see negative 25 plus 70 let's see negative 20 plus 70 would be negative would be positive 50 and then you have another 5 so it's plus 45 plus 45 so this we just we've just algebraically manipulated this equation and we get 0 is equal to X plus 5 squared plus 45 now we could have from the beginning if we wanted we could have tried to factor it but we're going to do here this will always work even if you have crazy decimal numbers here you can solve for X using the method we're doing here completing the square so to solve for X let's just subtract 45 from both sides of this equation let's subtract 45 and so the left-hand side of this equation becomes negative 45 negative 45 and the right hand side will be just the X plus 5 squared these guys right here cancel out now normally if I look at something like this I'll say okay let's just take the square root of both sides of this equation and so you might be tempted to take the square root of both sides of this equation but immediately when you do that you'll notice something strange we're trying to take the square root of a negative number and if we're dealing with real numbers which is everything we've dealt with so far you can't take a square root of a negative number there are no there is no real number that if you square it will give you a negative number so there's it is not possible I don't care what you make X it is not possible to add X to 5 and square it and get a negative number so there is no X there is no Exce that can satisfy can satisfy if we assume that X is a real number that can satisfy this equation because I don't care what you what X you put here what real X you put here you add 5 to it 2 squared there's no way you're going to get a negative number so there's no X that can satisfy this equation so there are we could say there are no and I'm using the word real because the in algebra to you'll learn that there there are things called complex numbers we're don't worry about right now but there are no real roots to the quadratic to the quadratic equation quadratic equation and we're done and actually if you would try to factor it you would have found it very difficult because this is not a factorable this is not a factorable expression right here and you know it because there's no real roots