 So once you know how to solve a quadratic equation, you can solve a wide variety of equations that are reducible to quadratics. This comes about in the following way. If we know how to solve x squared plus 5x plus 4 equals 0 for x, then we know how to solve an equation like blank squared plus 5 blank plus 4 equals 0 for blank. And that's because whatever process we did to solve this equation for x, we can apply exactly the same process to solve this equation for blank. And this leads to the following idea. We say that an equation is reducible to quadratic if we can, through some substitution of the variables, transform it into a quadratic equation. Well, let's see how that works. So let's solve x squared plus 5x plus 4 equals 0, and then let's solve 3z plus 1 squared plus 5 times 3z plus 1 plus 4 equals 0. So solving x squared plus 5x plus 4 equals 0, well, let's use that quadratic formula and we get our two solutions x equals negative 1 or x equals negative 4. What about the second equation? To solve the second equation, notice that both of these terms have a 3z plus 1 expression. And so this suggests we'll use the substitution x equals 3z plus 1 equals means replaceable. So every place we see a 3z plus 1, we'll replace it with x and get, but we already know the solutions to this equation equals means replaceable. So if x equals 3z plus 1, we can replace x with 3z plus 1 everywhere we see it. And so now we get two new equations in z. Solving each one of these separately gives us our solutions. Now the challenge here is coming up with the right substitution. And the bad news is there is no algorithm. There is no formula. There is no certain way that you can know what the exact substitution is going to be. All you can do is try a bunch of different things and see what works. But remember, it doesn't matter if you don't succeed at first. What matters is if you succeed at last, keep trying until you find something that works. So let's try to solve z to the fourth minus 5z squared plus 4 equals 0. Now a little analysis goes a long way. The thing that you might notice here is that the exponents on z are both even. They're 2 and they're 4. And that suggests that if x equals z squared, x squared, well, that's z squared squared, otherwise known as z to the fourth. And that gives us both of our variable expressions. So let's make that substitution x equals z squared and x squared equals z to the fourth. And remember, equals means replaceable. So here, any place I see z squared, I can replace it with x. And here, any place I see z to the fourth, I can replace it with x squared. So my equation, so if I make these replacements, I get x squared minus 5x plus 4 equals 0. Since you'd never be asked to solve a hard problem, you can trust that the equation we get is factorable. Well, in this case, we do find that we can factor this expression over on the left-hand side. It factors as, and since we have product equal to 0, we know that one of the factors must be 0. So either x minus 4 is equal to 0 or x minus 1 is equal to 0. Solving these equations give us our two solutions. If it's not written down, it didn't happen. So we'll write down those solutions. And squirrel, here's why it's important to write things down. It's very easy to get sidetracked while solving any sort of extended problem. Having everything written down makes it easier to come back to the problem. We want to solve this equation. What we found so far is that x equals 4 or x equals 1. But our original equation is in z, and we need to find the value of z. Because we've written down x equals z squared, equals means replaceable. So from x equals 4, x equals 1, we can replace and get our equations in z, which we can then solve. Since z squared equals 4, then by the square root property, we know that z is plus or minus the square root of 4. And that means that z is plus or minus 2. Likewise, since z squared equals 1, we know that z is plus or minus the square root of 1, which is plus or minus 1.