Video: Solving Quartic Equations

Solve the equation (𝑥 − 1)(𝑥 + 6)(𝑥 − 4)(𝑥 + 7) = 0.

02:29

Video Transcript

Solve the equation 𝑥 minus one multiplied by 𝑥 plus six multiplied by 𝑥 minus four multiplied by 𝑥 plus seven equals zero.

So you might think, well, what do we need to do? Do we need to start distributing across our parentheses. But in fact, no, cause what we have is an equation in fully factored form. So therefore, finding the solutions is fairly simple. Well, all we need to do is take a look at our parentheses and see which 𝑥-values are gonna make them equal to zero. And that’s because if one of them is equal to zero, that means the answer to the left-hand side is gonna be equal to zero because zero multiplied by anything is zero. And on the right-hand side of the equation, we have zero, and that wants to be our solution.

So what we can do is equate them to zero, so if we’ve got 𝑥 minus one equals zero, well, then the value of 𝑥 that’s gonna make that parentheses equal to zero is gonna be one. So that’s our first solution. Then for our second solution, what we’ve got is 𝑥 plus six equals zero. So then if we subtract six from each side of the equation, we’re gonna get 𝑥 equals negative six. So that’s our second solution. And then, for our third solution, we take a look at the parentheses where we’ve got 𝑥 minus four. And then if we equate that to equal zero, then we just add four to each side, so we get 𝑥 equals four.

So then we use the same method for our final solution, so we get 𝑥 equals negative seven. So therefore, we can say the solutions are 𝑥 equals one, 𝑥 equals negative six, 𝑥 equals four, and 𝑥 equals negative seven. And for a quick tip to solve a question like this nice and easily, you might’ve noticed that, in fact, the values of 𝑥 were just the opposite sign to the value that we had inside the parentheses. So we had plus one or positive one instead of negative one, negative six instead of positive six, positive four instead of negative four, and negative seven instead of positive seven.

And although this is a very nice and quick, easy way to solve a problem like this, it is worth mentioning it is slightly different if we have something like this. So here, we’ve got two 𝑥 plus three. So if this was our parentheses, then we have to equate this to zero. Well, if we’ve got two 𝑥 plus three equals zero and we subtract three from each side, we get two 𝑥 equals negative three. And then to work out what 𝑥 is, we divide by two. So, in fact, what we have is 𝑥 is equal to negative three over two. So it’s not just negative three; it’s negative three over two. And that’s because the coefficient of 𝑥 was greater than one. So be careful if you have something like this.

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