WEBVTT
00:00:01.890 --> 00:00:05.510
Factor Quadratics When 𝑎 Is Not Equal to One
00:00:07.340 --> 00:00:12.470
We know that any quadratic comes in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐.
00:00:12.820 --> 00:00:23.480
And to factor a quadratic when 𝑎 is not equal to one, we need to employ the first step of when it is equal to one, so we need to draw a table.
00:00:25.000 --> 00:00:35.750
And in this table, we are looking for something that our two factors will add to get and something that they will multiply to get, so we’re looking for the sum and the product.
00:00:36.190 --> 00:00:44.560
Well the sum will always be the number in front of the 𝑥, or we could say the 𝑥-coefficient.
00:00:45.140 --> 00:00:54.840
And then the product will always be 𝑎 multiplied by 𝑐 to the coefficient of 𝑥 squared multiplied by the number by itself.
00:00:56.510 --> 00:01:05.910
And then we’ll list the factor pairs of 𝑎𝑐 to find out which factor pair fits with our quadratic, so let’s have a go at an example.
00:01:07.890 --> 00:01:12.960
We must factor six 𝑥 squared minus five 𝑥 minus four.
00:01:13.380 --> 00:01:15.410
First thing we need to do, draw a table.
00:01:16.480 --> 00:01:21.990
So we can see that the sum will be the coefficient of the 𝑥, so that will be negative five.
00:01:23.240 --> 00:01:31.790
And the product will be six multiplied by negative four, which is equal to negative twenty-four.
00:01:32.660 --> 00:01:44.980
Now, the fact that our product is negative tells us that one of the factor pairs must be negative and one must be positive, and we’re looking for a factor pair that has a difference of five.
00:01:45.450 --> 00:01:49.560
So, what we now do is list the factor pairs of twenty-four.
00:01:50.610 --> 00:01:52.760
First of all, we’ve got one and twenty-four.
00:01:54.080 --> 00:01:56.150
They do not have a difference of five.
00:01:57.140 --> 00:01:59.950
Two and twelve, nope, they don’t either.
00:02:01.320 --> 00:02:04.960
Three and eight, aha, yes they do.
00:02:05.370 --> 00:02:13.480
So because we know that they add to give us a negative five, then that tells us that the larger number, eight, must be negative.
00:02:15.780 --> 00:02:23.160
And the reason we found the factor pair in this case is basically what we’re doing is we’re splitting that middle 𝑥-value.
00:02:25.090 --> 00:02:30.740
As we know, that three 𝑥 minus eight 𝑥 is negative five 𝑥.
00:02:32.070 --> 00:02:41.030
Now, we’re going to take each set of two terms separately and factor them like they are a linear expression.
00:02:41.660 --> 00:02:50.910
So, taking the first pair, we can see that they’ve both got three in common and also 𝑥.
00:02:50.940 --> 00:03:01.160
So if we factor that, then we know that three multiplied by two is six and 𝑥 multiplied by 𝑥 is 𝑥 squared.
00:03:02.340 --> 00:03:08.800
And then for the next term, we’re asking what do I have to multiply three 𝑥 by to get three 𝑥.
00:03:08.980 --> 00:03:09.860
Well that’s just one.
00:03:11.210 --> 00:03:15.690
Be careful not to miss that one because that’s something that students sometimes do.
00:03:16.720 --> 00:03:18.670
Now let’s take the next pair.
00:03:20.260 --> 00:03:27.100
We can see that they have both got a common factor, a greatest common factor, of negative four.
00:03:28.810 --> 00:03:36.690
Now, this is a moment to be careful because us- each parenthesis must be exactly the same inside.
00:03:37.490 --> 00:03:42.890
So negative eight 𝑥 divided by negative four is two 𝑥.
00:03:43.750 --> 00:03:49.220
That’s okay so far, and then negative four divided by negative four is one.
00:03:49.690 --> 00:03:50.620
Here we go; we’re okay.
00:03:51.840 --> 00:04:07.550
The reason that these must be the same is because, now, if we look at that expression, we can see that each term actually has a factor of two 𝑥 plus one, so we’re going to factor two 𝑥 plus one.
00:04:08.880 --> 00:04:14.250
And that will give us our first set of parentheses, and then we look at what’s left over.
00:04:14.310 --> 00:04:17.720
We can see we’ve got a three 𝑥 and a negative four.
00:04:18.950 --> 00:04:23.230
So our second parenthesis becomes three 𝑥 minus four.
00:04:24.400 --> 00:04:25.230
Now we have it.
00:04:27.210 --> 00:04:35.470
So just going back over what we did, first of all, we need to find what the sum and what the product are.
00:04:36.030 --> 00:04:43.130
So in this case, the sum was negative five and the product was six times negative four, so negative twenty-four.
00:04:43.730 --> 00:04:52.660
We then needed to list the factor pairs of twenty-four to try and find two numbers that would satisfy this sum and product.
00:04:53.540 --> 00:05:02.240
The reason we were doing this was because we were splitting our middle 𝑥-value, giving us three 𝑥 and minus eight 𝑥.
00:05:03.080 --> 00:05:18.310
Then we took each pair of terms individually and factored them and then factored the same parentheses, giving us two 𝑥 plus one all multiplied by all of three 𝑥 minus four.