Factoring Using the Texas Method
For the Texas Method, you’ll draw a capital T and X. Place the first term on top of the T. At the top of the X, multiply the first coefficient by the constant. In the bottom of the X, place the middle coefficient. On either side of the X, place the two numbers that have a product of the first coefficient and sum of the bottom coefficient. These two numbers also go on either side of the T, and need to be simplified. Once you’ve checked your factoring, use the zero product property to solve the equation.
Factoring Using the Texas Method
Solve the equation 2x² minus 5x, minus 12, equals 0 by factoring. There are many different ways to factor, I’m going to show you one of my favorites which is called the Texas Method. It’s called the Texas method because you’re going to draw a “T” and an “X.”
On top of the “T,” you’re going to take the first term, 2x², and you’re going to put 2x and 2x. Notice the ² is missing, that’s because we’re looking over here for what’s going to go into our binomials.
I know it can’t be to 2x² and 2x², because that would give me 4x^4, so it’s just 2x and 2x, and we’re going to simplify at least one of them, possibly both, in just a minute; but you know what, before you do any of that you should look to see if there’s a GCF. Always, the first thing you want to do when you’re going to factor something is see if there’s a greatest common factor.
In this case there isn’t, so moving on with the Texas Method. Okay, in the top of your “X” you’re going to multiply the first coefficient (or your leading coefficient) times the constant, and sometimes your leading coefficient will be 1.
Like if there wasn’t a 2 here, well then you would just do 1 times negative 12, but this time there is a leading coefficient, so 2 times negative 12 is negative 24. In the bottom of the “X” is your middle term, which is negative 5 (or your middle coefficient really because we’re leaving off the x).
Now on either side of the “X,” we’re looking for those two numbers that would multiply to give us negative 24 and add to give us negative 5, two numbers that have a product of negative 24 and a sum of negative 5. Those numbers would be negative 8 and positive 3, negative 8 times 3 is negative 24, and negative 8 plus 3 is negative 5, so those two numbers go in our “T,” negative 8 and 3.
We’re almost done, but we know that our two binomials couldn’t be 2x minus 8 and 2x plus 3, because we’d have 2x times 2x, which would give us 4x² when we FOILed it out. The next thing we’re going to do is simplify, and we can simplify, and you just simplify like you would with a fraction, so if you think of this as just 2/8, then you could simplify that to 1/4.
Divide by 2 and you get just x, divide by 2 and this is going to be a negative 4 instead, so we have x minus 4, and then our other, 2x and 3, can’t be simplified, so it’s just 2x plus 3, and, of course, it’s still equal to 0. Let me bring this down here where we have more space, x minus 4, times, 2x plus 3, equals 0.
Of course, any time you factor you should always do a quick check to make sure that you didn’t mix up a plus and a minus or a mess up a number somewhere, so I like to just do a quick FOIL. X times 2x, yep, 2x². X times 3, 3x. Then, negative 4 times 2x, that’s negative 8x plus 3x does give us the negative 5x.
Finally, negative 4 times 3, that’s the negative 12. Now that I’ve confirmed that I did factor correctly, we can use the Zero Product Property to solve, which states that if the product of two expressions is 0, one of those expressions must be 0.
Either x minus 4 is 0, or 2x plus 3 is 0, but if you’re going to multiply two things together and get a 0, one of those things has to be 0. Add 4 to both sides to solve for x here, so x is 4, subtract 3 from both sides, 2x equals negative 3, then divide both sides by 2 and you get that x is negative 3/2, or x is negative 1 and 1/2, so x is negative 1 and 1/2 and 4, those are our two answers. Negative 1 and 1/2 and 4 would cause this expression to equal 0.