Hi guys! I'm Nancy
and I'm going to show you
how to solve a quadratic equation by factoring.
OK. First we're going to look at a simpler one
and just use a brute force method.
And then I'm going to show you a tougher one
where it's going to be faster and easier
to use a trick called the "magic x" method
or "x method"
and that's for when the brute force method would take way too long.
And this trick will absolutely give you your factors
so that you can solve your equation.
OK. So say you have this quadratic equation
that starts with x^2 and you need to solve it by factoring.
OK, so what you want is two factors
each starting with x
that multiply to your expression and equal zero.
So what you need to do to factor
is find two numbers
that multiply to -16
(your last number)
and add to positive 6.
(your second number)
So that's two numbers that multiply
to -16 and also
add to your middle number, 6.
So again, we're looking for two numbers that multiply
to -16 and add to positive 6.
So first what you need to do is make a list
of all pairs of numbers that multiply to -16.
And you can do that off to the side.
We want two numbers that multiply to -16.
Let's list the pairs.
We could have 1 and -16.
We could have flipped signs. -1 and 16.
We could have 2 and -8.
Flip the signs. -2 and 8.
We could also have 4 and -4.
Or the opposite. -4 and 4.
OK, so you've made a list of all numbers
that multiply to -16.
Now let's look at the other requirement.
We need to find which ones add to positive 6.
So let's check them.
1 + -16 would add to some negative number.
So that can't be it.
-1 + 16 would give you positive 15.
It's a bit too large.
2 + -8 would give you -6.
It's close, but it's the wrong sign.
-2 + 8 would add to positive 6.
So those are your factor numbers.
Now you don't need to check any pairs that are left.
You can assume they're not right.
You've found your factor numbers, -2 and 8.
OK, so now you've found your two numbers
and you just need to fill them in, into your factors.
So you found -2 and 8.
So you simply put -2 here
positive 8 there
and you found your factors.
So by the way, you could also switch these numbers
you could change the order and it would be fine.
+8, -2
because these are both "x +" in your factor terms.
The order actually wouldn't matter.
OK, so this can actually just be simplified to:
(x - 2) (x + 8)
equals zero.
OK, so now you've factored and you're ready to solve
and all you need to do
is take each of these factors
and set them each, separately, equal to zero.
So now pull each of these out
and write them as a separate equation.
We have (x - 2) set equal to 0.
x - 2 = 0
Use your algebra skills to solve
you add 2 to both sides
so that you just have: x = 2
So there's one of your solutions: x = 2.
OK, so we'll do the same thing for the other factor.
Pull it out separately.
You have (x + 8) set equal to zero.
So that's: x + 8 = 0
Solve by subtracting 8 from both sides.
Minus 8. Minus 8.
And we just have x alone equals -8.
So that is your other solution.
So x = 2 and x = -8 are your two solutions
that you found by factoring your quadratic.
OK. Say you have a tougher example
Where your quadratic equation
starts with a term like 2x^2 instead of just x^2.
So what you have to do first
is check to see if you can factor out a number overall.
OK, so in this problem
we can actually pull out a 2 from every term.
So we would have 2
parenthesis
and then what's left over is x^2
plus 6x
minus 16
equals zero.
So notice that this problem
was actually just like the last one, but hidden.
It was disguised by an overall 2 constant.
So this would actually just factor
using the same brute force, simpler method
from the last problem.
And you would have
2 times (x - 2)
(x + 8)
So watch out for an overall constant
that's making it look harder than it actually is.
OK. Say you have a tougher quadratic equation
that starts with 3x^2
and you want to solve by factoring.
So you need to find two factors
so that your equation looks like this
and equals zero.
OK, so the first thing you do is literally
just draw an X off to the side.
So in the top of your X
you're going to put the number you get
from multiplying this first coefficient, 3,
by your last number, -5.
So that's 3 times -5.
And that's going to go in the top of your X.
So we have -15 in the top of your X.
Then, in the bottom of your X
you're just going to put your second constant
your second coefficient.
So we put positive 14 in the bottom of the X.
Now here's the trick
you want to find two numbers
that multiply to -15
and add to 14.
So that's two numbers that multiply
to -15.
And.. add..
to positive 14.
So let's list all the pairs of numbers
that multiply to -15.
Now, test to see which of these pairs
add to positive 14.
1 and -15 doesn't.
-1 and 15 adds to positive 14.
So those are your two numbers.
Now let's write -1 and 15
on either side of your X.
We have -1
and 15.
It doesn't matter what order you put those in.
You can flip the order.
Now here's an important step.
You're going to
divide each of those two numbers
by your leading coefficient
which in this problem is 3.
So we're going to divide -1 by 3
and divide 15 by 3.
Now that simplifies.
So your simplified X would have
the same top and bottom numbers
-1/3.
And 15/3 simplifies to just
5/1.
OK, you're almost done, you're very close to factoring.
You're going to take these two fractions
and use them to write your factors.
So the bottom number here.. 3
is your coefficient of x.
And your top number, -1
is your constant. So just write -1.
Same thing goes for the other factor
where you use this fraction.
Your bottom number is your coefficient of x.
And your top number is just your constant.
So plus 5.
So you have a factored quadratic equation
and you're ready to solve.
So what you do is take each factor
and separately set them equal to zero.
So this one equal to zero.
And this set of parentheses equal to zero.
Now let's pull that out separately down below.
So we have (3x - 1) set equal to zero.
As its own equation that's:
3x - 1 = 0
And when we solve that we get:
3x = 1
or x = 1/3
So that's one of your solutions.
Now we'll do the exact same thing for the other factor.
Pull it out down below so we have
1x + 5 set equal to zero.
And when you solve that
you will get: x = -5
And that is your other solution.
So now your two answers, your two solutions are
1/3 and -5
and you found them by factoring your quadratic equation.
So I hope that trick helped save you some time.
And if it did
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