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# Solving quadratics using structure

## Video transcript

so let's try to find the solutions to this equation right over here we have the quantity 2x minus 3 squared and that is equal to 4x minus 6 I encourage you to pause the video and give it a shot and I'll give you a little bit of a hint you could do this in the traditional way of expanding this out but there might and then turning it into a kind of a classic quadratic form but there might be a faster or a simpler way to do this if you really pay attention to the structure of both sides of this equation well let's look at this we have 2x minus 3 squared on the left hand side on the right hand side we have 4x minus 6 well 4x minus 6 that's just 2 times 2x minus 3 let me be clear there so this is the same thing as 2x minus 3 squared is equal to 4x minus 6 if i factor out a 2 that's 2 times 2x minus 3 and so this is really interesting we have something squared is equal to 2 times that something so if we can solve for the something let me be very clear here so the stuff in blue squared is equal to 2 times the stuff in blue so if we can solve for what the stuff in blue could be equal to then we could solve for X and I'll show you that right now so let's say let's just replace 2x minus 3 we'll do a little bit of a substitution let's replace that with let's replace it with P so let's say that P P is equal to 2x minus 3 well then this equation simplifies quite nicely the left hand side becomes P squared P squared is equal to 2 times 2 times P because once again 2x minus 3 is P 2 times P and now we just have to solve for P and I'll switch to just one color now so we can write this as if we subtract 2 P from both sides we can get P squared minus 2 P is equal to 0 and we can factor out a P so we get P times P minus 2 is equal to 0 and we've seen this show multiple times if I have the product of two things in the equal to zero at least one of them needs to be equal to zero so either P is equal to 0 or P minus 2 is equal to 0 well if P minus 2 is equal to 0 that means P is equal to 2 so either P equals 0 or P equals 2 well we're not quite done yet because we wanted to solve for X not for P but luckily we know that 2x minus 3 is equal to P so now we could say either either 2x minus 3 is going to be equal to this p value is going to be equal to 0 or 2x minus 3 is going to be equal to this p value is going to be equal to 2 and so this is pretty straightforward to solve add 3 to both sides you get 2x is equal to 3 divide both sides by 2 and we get X is equal to 3 halves or over here if we add 3 to both sides we get 2x is equal to 5 divide both sides by 2 and you get X is equal to 5 halves so these are the possible solutions and this is pretty neat this one right over here you can almost do this in your head it was nice and simple well if you were to expand this out and then subtract this it would have been a much more complex set of operations that you would have done you still would have hopefully gotten to the right answer however just taking a lot more steps but here we could appreciate some patterns that we saw in our equations and namely we have this thing being squared and then we have 2 times that same thing 2 times 2x minus 3