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How to Solve Quadratic Equations by Factoring (NancyPi)

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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