Most Students Struggle With This SAT Algebra Problem | Can You Solve it?

Added: 2026-05-09

[00:00:00]Hello and welcome. There's a very simple way to solve this equation without first of all having to multiply all these four factors and get a quatic equation. Let us start by rearranging these factors.

[00:00:15]Now note that 1 * 6 is equal to 6 and also 2 * 3 is equal to 6. So let us write this first. We have x + 2 followed by this x + 3 then followed by this x + 1 and lastly x + 6 and of course this is equal to 3 x 2. Now let us multiply these two factors. We have x * x which is x^2.

[00:00:51]Then 2 + 3 is going to give us 5. So we have + 5 x. Then 2 * 3 is of course 6.

[00:01:04]Now let us multiply these two factors.

[00:01:07]We have x * x which is x 2. 1 + 6 is 7.

[00:01:14]So we have 7 x and 1 * 6 is 6. So this is 6. And of course this is still equal to 3x^2.

[00:01:26]Now here we can write this as x^2 + 6 x but you can see that we have 5x here we have + 6 that is this. What do we do to 6x to get 5x? Of course we subtract 1 x.

[00:01:45]Now in this case we have x^2 + 6 x + 6.

[00:01:52]Now what do we do to 6 x to get 7 x? We add 1 x.

[00:01:59]And of course this is still equal to 3x^2.

[00:02:03]Now notice that here we have x^2 + 6 x + 6 - x. And here we have x^2 + 6 x + 6 + x. This looks something like a - b ult* by a + b where a is x^2 + 6 x + 6 and of course b is x and you know that this is the difference of two squares.

[00:02:35]when we multiply these two what we have is a^ 2 - b².

[00:02:43]So that simply means that when we multiply these two factors what we are going to get is a 2 that is x 2 + 6 x + 6 2 - b 2 which of course is x 2 and this is still equal to 3 x^2.

[00:03:12]And now let us move 3x^2 over to the left hand side of the equation. When we do that we have x^2 + 6 x + 6 2 - x^ 2 - 3 x^2 is = 0.

[00:03:32]- x^2 - 3x^2 is going to give us -4 x^2.

[00:03:39]we have x2 + 6 x + 6 2 and this is equal to zero. But of course you know that we can write 4 x^2 as 2 x^ 2. We still have x 2 + 6 x + 6 2. Once more look at this.

[00:04:06]Of course this is equal to zero.

[00:04:09]You can see the difference of two squares. And you know how we can factoriize this left hand side? We can factoriize this as x^2 + 6 x + 6 + 2 x * x^2 + 6 x + 6 - 2 x and of course this is equal to zero.

[00:04:43]6 x + 2 x is 8 x. So here we have x^2 + 8 x + 6.

[00:04:55]6 x - 2x is 4x. Here we have x^2 + 4 x + 6. And of course this is equal to zero.

[00:05:08]Now you can see that we have succeeded in factorizing the left hand side of this equation. And of course you know that when the product of two items is equal to zero then it simply means that either one of the items must be equal to zero. So from here we have that either x^2 + 8 x + 6 is = 0 or x 2 + 4 x + 6 is equal to 0.

[00:05:46]Now we cannot solve this quadratic equation by factorization. So let us solve it using the completing the square method. Since the coefficient of x squ is already one, the next thing we do is to move the constant term over to the right hand side. When we do that, we have x^2 + 8 x is equal to -6. Now the next step is to divide the coefficient of x by 2, square it, and add to both sides of the equation to complete the square on the left hand side. When we do that we have x^2 + 8 x + half of 8 is 4.

[00:06:27]So we have 4^ 2. This is = -6 + 4^ 2 is 16. Now this is a perfect square. So we take 1 x and we take 1 4 and we square.

[00:06:47]This is equal to 16 - 6 which is equal to 10. And of course the next thing we are going to do is to take square root of both sides of this equation. When we do that we have that x + 4 is equal to plus or minus the square root of 10. Now let us subtract four from both sides of this equation to find the values of x.

[00:07:11]From here we have that x is = -4 + or minus the square roo<unk> of 10.

[00:07:20]Now once more we cannot solve this equation by factorization. So let us solve it by completing the square method. The coefficient of x² is already one. So we move the constant term over to the right hand side. When we do that, we have x^2 + 4x is = -6.

[00:07:43]Once more, we divide the coefficient of x by two, square it, and add to both sides of the equation. When we do that, we have x 2 + 4 x + 2. And of course, this is equal to -6 + 2^ 2.

[00:08:04]This is a perfect square. So we take one x and we take one 2 and we square. This is equal to 2^ 2 is 4. 4 - 6 is -2. So on the right hand side we have -2.

[00:08:25]And finally we need to take square root of both sides of this equation. When we do that, we have that x + 2 is = plus or minus the square roo<unk> of -2. And of course, you know that this is the same thing as plus or minus the square<unk> of -1 * 2.

[00:08:48]Now we have that x + 2 is = + or minus the square root of -1 is i. Then we have roo<unk>2.

[00:09:01]And finally to find the values of x we subtract two from both sides of this equation. When we do that we have that x is = - 2 + or minus i<unk> 2. So we have all four values of x that satisfy this algebraic equation. x = -4 +<unk> 10. x = -4 -<unk> 10 x = -2 + i <unk>2 and x = -2 - i <unk>2. Obviously you can see that two of the solutions are complex while two of the solutions are real.

[00:09:47]And with that you come to the end of this tutorial. I hope you learned something new. If you enjoy such content, please subscribe to the channel. Leave us a like to support the channel. Thanks for watching and you can see more tutorials here.