Solving quadratics using structure | Mathematics II | High School Math | Khan Academy

So let's try to find the solutions to this equation right over here. We have the quantity (2x - 3) squared, and that is equal to (4x - 6). I encourage you to pause the video and give it a shot. I'll give you a little bit of a hint: You could do this in the traditional way of expanding this out and then turning it into 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 - 3) squared on the left-hand side. On the right-hand side, we have (4x - 6). Well, (4x - 6) that's just (2 \times (2x - 3)). Let me be clear there. So this is the same thing as (2x - 3) squared is equal to (4x - 6). If I factor out a two, that's (2 \times (2x - 3)).

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 two times the stuff in blue. 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 - 3)—we'll do a little bit of a substitution. Let's replace that with (P). So let's say that (P) is equal to (2x - 3). Well then this equation simplifies quite nicely. The left-hand side becomes (P^2), and (P^2) is equal to (2 \times 2 \times P) because once again, (2x - 3) is (P), (2 \times P).

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 (2P) from both sides, we get (P^2 - 2P) is equal to zero. We can factor out a (P), so we get (P(P - 2)) is equal to zero.

And we've seen this show multiple times. If I have the product of two things and they equal to zero, at least one of them needs to be equal to zero. So either (P) is equal to zero or (P - 2) is equal to zero. Well, if (P - 2) is equal to zero, then that means (P) is equal to (2). So either (P = 0) or (P = 2).

Well, we're not quite done yet because we wanted to solve for (x), not for (P). But luckily, we know that (2x - 3) is equal to (P). So now we could say either: either (2x - 3) is going to be equal to this (P) value, which is equal to zero, or (2x - 3) is going to be equal to this (P) value, which is equal to two.

And so this is pretty straightforward to solve. Add three to both sides, you get (2x) is equal to three. Divide both sides by two and we get (x) is equal to (\frac{3}{2}).

Or over here, if we add three to both sides, we get (2x) is equal to five. Divide both sides by two and you get (x) is equal to (\frac{5}{2}).

So these are the possible solutions, and this is pretty neat. This one right over here you could almost do this in your head. It was nice and simple. While 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, but it would have just taken a lot more steps.

But here we could appreciate some patterns that we saw in our equations, namely we have this thing being squared and then we have two times that same thing (2 \times (2x - 3)).