These factorization tricks will blow your mind

Added: 2026-05-21

[00:00:00]Hello and welcome. In this math tutorial, our task is to solve this quatic equation for real numbers. Now, this equation does not have rational roots. But luckily for us, there are several very simple ways of solving the equation using factorization methods.

[00:00:18]Now, notice that this equation does not have any terms in x². So one way of solving it by factorization is to add and subtract 4x^2 to the left hand side of the equation. Now when we do that we have x raised to power 4 + 3 x cub + 4 x^2 - 4 x^ 2 + 6 x + 4 and of course this is equal to zero. Now let us reorder the terms. Let us take this this and that. We have x ra to power 4 + 4 x 2 + 4. Now we take this and that we have 3 x cub + 6 x and finally we are left with this. So we have - 4x^2 and this is equal to 0. Now look at these first three terms. We can factoriize them.

[00:01:34]What are the factors of x raised to power 4? We have x^2 and x 2. What are the factors of 4? Of course 2 * 2. x^2 * 2 will give us 2 x^2 and in the same way x^2 * 2 will give us 2 x². I am sure you already know where this is going. Now when we factoriize these three we have x 2 + 2 * x^2 + 2 which is exactly the same thing. So this is squared. Now look at these two terms. You will see that they both have a common factor which is 3 x.

[00:02:19]So we have + 3 x into now 3x cub / 3x is going to give us x² and of course 6 x / 3x is going to give us 2. Finally we are left with -4 x^2 and this is equal to zero.

[00:02:44]Now notice that here we have x^2 + 2 and here we also have x2 + 2. So let x2 + 2 be equal to the letter d. That means that we can write this equation as d^ 2 + 3 x dus 4 x^2 is equal to 0. Now we can factoriize the left hand side of this equation. What are the factors of d ^ 2?

[00:03:20]We have d * d. What are the factors of - 4x^ 2? we have 4 x * - x. Now d * - x is going to give us - x d and d * 4x is going to give us 4 x d. And of course notice that 4x d minus xd is going to give us 3 xd. So when we factoriize this we have d + 4 x * d - x and of course this is equal to zero. But remember that d is = x^2 + 2.

[00:04:08]So here we have that x^2 + 4 x + 2 * x^2 - x + 2 is equal to zero. So you can see how we have successfully factorized the left hand side of this quadic equation. So to find all the values of x that satisfy this equation, all we need to do is to solve these two equations.

[00:04:41]Now an alternative way of factorizing this quatic equation is to divide through the equation by x².

[00:04:51]And of course you know that we can do this because we know that x=0 is not a root of this quic equation because if we substitute x= 0 then we have + 4. Now when we do the division, this divided by that is x².

[00:05:10]This divided by that is 3 x.

[00:05:15]This divided by that is 6 / x. And this divided by x 2 is 4 / x². And of course this is equal to zero. Now let us collect like terms. We have x^2 + remember that 4 is 2^ 2 divided by x^ 2 then we have + 3 x + 6 / x.

[00:05:53]Now this is equal to zero. Of course, we can write this as x^2 + 2 / x^ 2 + Now, when you look at these two terms, you're going to see a common factor, which is three. So, we have 3 into x + 6 /x / 3 is 2 / x. This is equal to zero.

[00:06:23]Now let us look at these two terms.

[00:06:26]Notice that here we have something that looks like a 2 + b 2. Of course a is x while b is 2 / x. Now remember that we can also write a 2 + b 2 as a + b 2 - 2 * a * b. So that means that we can write this as x + 2 / x^ 2 - 2 * x * 2 / x. And then of course we have + 3 * x + 2 / x and this is equal to zero. Now notice that here we have x + 2 /x and also here we have x + 2 /x. So let x + 2 /x be equal to the letter d. That means that we can write this equation as d^ 2 - 2 * this is going to cancel that 2 * 2 is equal to 4. So here we have 4 then + 3 d and this is equal to zero.

[00:08:04]Now rearranging this quadratic equation we have d^ 2 + 3 d - 4 is equal to zero.

[00:08:14]And of course you know that we can factoriize the left hand side of this equation. Since the coefficient of d² is 1. All we need to do is to find the factors of -4 that add up to + 3. And of course we have 4 and -1 because 4 * -1 is going to give us -4 while 4 -1 is equal to 3. Now when we factoriize this we have d + 4 * d - 1 is equal to zero. And from here it's very easy to see that either d + 4 is equal to zero or d minus1 is equal to zero. Here we have that d is equal to -4 while here we have that d is equal to 1. But now remember that x + 2 /x is equal to d. So this simply means that x + 2 /x is equal to -4. Now to get rid of this fraction, we multiply through this equation by x. x * x is x^ 2. x * 2 / x is 2. And of course - 4 * x is -4 x. Rearranging this equation, we have x^2 + 4 x + 2 is equal to zero. In this case, we have that x + 2 / x is equal to 1. Once more, to get rid of this fraction, we multiply through this equation by x. x * x is x 2. 2 / x * x is 2. and 1 * x is = x. Rearranging this equation, we have x^2 - x + 2 is equal to zero. Once more, you can see that we have arrived at the exact same two quadratic equations. x^2 + 4 x + 2 is = 0 and x^2 - x + 2 is equal to zero. And finally, let us solve the two quadratic equations that we obtained. In this case, we can't solve this equation by factorization. So, let us solve it using the completing the square method. Since the coefficient of x^2 is 1 already, the next thing we do is to move the constant term over to the right. When we do that, we have x^2 + 4x is equal to -2. Now let us divide the coefficient of x which is 4 by 2. Square it and add to both sides of the equation to complete the square. On the left hand side we have x^2 + 4 x + 2 2 is = - 2 + of course 2^ 2 is 4. Now this is a perfect square.

[00:11:38]So we take one x and we take 1 2 and we square and this is equal to 4 - 2 is 2.

[00:11:48]Now we take square root of both sides of the equation to get that x + 2 is equal to + or minus <unk>2.

[00:11:58]And of course when we subtract two from both sides of this equation we have that x is equal to - 2 + or minus<unk> 2. Now in this case I do not think that this quadratic equation is going to yield real values of x but let us check using its discriminant. Now the discriminant is b ^ 2 - 4 a c b 2 is -1 2 which is 1 - 4 a is 1 and c is 2.

[00:12:38]we have 1 - 8 which is equal to -7. And of course you know that when the discriminant of a quadratic equation is negative it simply means that the quadratic equation has no real roots. So there are no real values of x that we can get from this quadratic equation. So the only two real values of x that satisfy this quotic equation are x = -2 + <unk>2 and x = -2 -<unk> 2.

[00:13:15]And with that we come to the end of this tutorial. I hope you learned something new. If you enjoy such content, please subscribe to the channel. Give us a like to support the channel. Thanks for watching and you can see more tutorials here.