so what are the five methods of solving quadratic equations these are the ones we're going to talk about in this video the graphing method quadratic formula factoring completing the square and taking square roots and more importantly I'm going to show you when you might want to use one method over another in a given problem so the first one I want to show you is this one x squared minus 2x minus 3 equals zero this one here we're going to show you the graphing method first okay and what you want to do is you want to look at y equals this equation here you want to get everything on one side you want to set it equal to zero and what we're going to do is we're going to graph this quadratic this parabola and we're going to look at where it crosses the x axis when it crosses the x axis that's where the Y value is equal to zero but the question is how do we graph it right well you can use this formula to find the vertex it's negative B divided by 2a so this is a B and C so the opposite of B is going to be positive 2 over 2 times a which you can see is 1 2 divided by 2 is 1 that's the x-coordinate of our vertex if we put it back in a 1 minus 2 is negative 1 minus 3 is negative 4 so you can see our vertex where the graph bends is that one negative 4 right there this is our axis of symmetry this is what divides our parabola in half and we can pick some points on either side of that axis of symmetry to get a better grasp so for example we put 0 in for X you can see that Y is going to be negative 3 right here and we can do reflected over that line of symmetry we can also pick let's say for example positive 3 excuse me that would be 9 minus 6 minus 3 which is 0 so here you can see we're at 0 and if we reflect that over that line of symmetry you can see we're gonna be over here at negative 1 so if we draw our parabola okay we can see where does it equal 0 where it's crossing right here at negative 1 and positive 3 those are going to be our two solutions so one would you want to use this method the only time I would really use this method as if I had a graphing calculator available or if they're giving you the graph already and you can just look at the x-intercepts that's when I would use the graphing method usually this is not the best method because it you know it takes a little bit more time and it doesn't necessarily guarantee that when you grow it when you are given a particular quadratic that's going to cross that nice integer values right so that's why we're gonna use these other tech so number two let's talk about using the quadratic formula so you probably ready know the quadratic formula but just a quick refresher it's x equals negative b plus or minus the square root of b squared minus 4ac all divided by 2a and that's when the quadratic is in this form ax squared plus BX plus C equals zero so you want to get everything on one side put it in descending order and set it equal to zero so that you can identify your a B and C values to put into that quadratic formula so let's go ahead and do this one and then I'll explain kind of why you would want to use a quadratic formula in a particular instance so here you can see the opposite of BC negative B is the opposite of B that would be positive 2 plus or minus negative 2 squared which is 4 minus 4 times a which is 3 times C which is negative 4 make sure you capture the sign if it's minus it's negative all divided by 2 times a which is going to be 6 so if we simplify this down a little bit further we have let's see negative 4 times 3 is negative 12 times negative 4 is 48 plus the 4 is 52 so we have 2 plus or minus square root of 52 all over 6 okay and what is the square root of 52 well 52 is really like 4 times 13 and we know the square root of 4 is 2 so we can simplify this down to 2 plus or minus 2 or 13 all over 6 and we can divide all these by 2 so this would be 1 plus or minus square root of 13 over 3 or you could write it separately as 1 plus square root of 13 over 3 and then also 1 minus square root of 13 over 3 and again all this is telling us is if we put this value here back in for X that's going to equal 0 or if we were to graph this parabola or this quadratic these are the values on the x-axis where the parabola would cross now why would we want to use the quadratic formula well you can see that this is not going to be easily factored this is also not going to be easily graphed and so when you have something like this it's not easy to factor or not easily graphed you know it's easy just to go ahead and jump right into the quadratic formula and you know get your exact values or you can put these in your calculator get an approximate value okay for number three now when we look at this one we're gonna use the factoring technique and the factoring technique I would only use of course when it's easily factored and when is it easy factored well a lot of times when you have a trinomial and the leading coefficients one the loans are easily factored you just have to say what multiplies to negative four what two numbers multiply to negative four but add to negative three that's going to be negative four and positive one so now if we set each group each factor equal to zero X minus four equals zero so that means that x equals 4 or X plus one equals zero which means that x equals negative one so again these are the two points you know where this parabola this quadratic would cross the x-axis so factoring is an easy method if it's easily factored okay for number four or x squared minus 8x minus six equals zero for this one we're going to do the completing the square method and when you complete the square what you want to do is you want to get the variables on the Left you want to move this constant over to the right side so what that gives us is x squared minus 8x equals six okay and so now what you want to do is you want to take half of this coefficient half of negative eight so we'll do negative 8 divided by 2 squared so that's going to be negative 4 the quantity squared which is 16 so what we're going to do is we're going to add 16 to the left side 16 to the right side to keep the equation balanced this is going to give us 22 and now what we've done by completing the square is we this is actually a perfect square you take half of this middle coefficient X minus 4 the quantity squared if this is plus a Texas would be plus 4 and so now do we have it in this form we can then solve by taking the square root of both sides remember when you do that you get two answers plus or minus and the square in this cancel so you just get X minus 4 and then if I add 4 to both sides we get x equals 4 plus or minus square root of 22 and you can write this separately 4 plus square root of 22 and 4 minus square root of 22 you can also do this on your calculator to get an approximate answer but the question is is when would you want to use the completing the square method well I would use this method when this middle coefficient this B value is even because if it was odd when we take half of that seven over two and we square it we're going to end up with a fraction and it's not going to be so easy to complete the square it's going to be you're gonna have a lot of fractions and there's going to be a little bit more complicated in that case I would probably just go and use the quadratic formula if the leading coefficients not one like say this was a two I would first divide everything by two it's easier to complete the square when that leading coefficient is one so if you do have that leading coefficient just divide it out the last example number five 4/9 x squared equals 12 we're going to use the taking square roots method and just like when we got down to this point here with completing the square we had a perfect square and we took the square root you know that's a good instance when you'd want to use a square root method but also what you can recognize when to use the square root method is when you have just like an x squared but not an X to the first okay so you just have a square term but not like a linear term and X to the first term so what we're gonna do is we're just going to get this X square by itself and the easiest way to do that is to multiply both sides by not 4/9 but the reciprocal nine force 12 is like twelve over one we can do a little bit of cross reducing three times 9 is 27 so these cancel and we get x squared equals 27 now we can take the square root of both sides remember when you do that you get two answers plus or minus 27 is really like 9 times 3 right and I'd have 3 is 27 square root of 9 that's a perfect square and we have this square root of 3 left over so it's going to be positive root 3 or positive 3 - square root 3 or negative 3 square root of 3 and again this is where the quadratic is crossing the x axis if you're graphing that parabola or it's if you take these values and put it back into the original equation it's going to make the equation true so those are the five techniques I'll put links in the description below two other videos I did talking about these different techniques if you want to learn more subscribe to the channel my goal for this channel is to make learning math less stressful so you can raise your grade pass your class and go on to pursue your dreams I look forward to helping in the future videos I'll talk to you soon