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Let's solve some quadratic equations by factoring.

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So let's say I had x squared plus 4x is equal to 21.

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Now your impulse might be to try to factor out an x and

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somehow set that equal to 21.

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And that will not lead you to good solutions.

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You'll probably end up doing

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something that's not justified.

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What you need to do here is put the entire quadratic

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expression on one side of the equation.

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We'll do it onto the left-hand side.

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So let's subtract 21 from both sides of this equation.

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The left-hand side then becomes x squared

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plus 4x minus 21.

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And then the right-hand side will be equal to 0.

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And the way you want to solve this, this

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is a quadratic equation.

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We have a quadratic expression being set equal to 0.

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The way you want to solve this is you want to factor them,

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and say, OK, each of those factors could

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then be equal to 0.

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So how do we factor this?

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Well, we saw in the last video that we have to figure out two

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numbers whose product is equal to negative 21, and whose sum

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is equal to 4.

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This would be a plus b would have to be equal to 4.

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Since their product is negative, they have to be of

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different signs.

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And so let's see, the number that jumps out

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at me is 7 and 3.

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If I have negative 7 and positive 3, I would get

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negative 4.

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So let's do positive 7 and negative 3.

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So the a and b are positive 7 and negative 3.

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When I take the product, I get negative 21.

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When I take their sum, I get positive 4.

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So I can rewrite this equation here.

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I could rewrite it as x plus 7, times x minus

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3, is equal to 0.

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And now I can solve this by saying, look, I have two

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quantities.

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Their product is equal to 0.

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That means that one or both of them have to be equal to 0.

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So that means that x plus 7 is equal to 0.

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That's an x.

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Or x minus 3 is equal to 0.

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I could subtract 7 from both sides of this equation.

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And I would get x is equal to negative 7.

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And over here, I can add 3 to both sides of this equation.

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And I'll get x is equal to 3.

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So both of these numbers are solutions to this equation.

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You could try it out.

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If you do 7-- negative 7 squared is 49.

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Negative 7 times 4 is minus 28, or negative 28.

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And that does indeed equal 21.

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And I'll let you try it out with the positive 3.

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Actually, let's just do it.

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3 squared is 9, plus 4 times 3 is 12.

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9 plus 12 is, indeed, 21.

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Let's do a bunch more examples.

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Let's say I have x squared plus 49 is equal to 14x.

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Once again, whenever you see anything like this, get all of

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your terms on one side of the equation and get a 0 on the

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other side.

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That's the best way to solve a quadratic equation.

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So let's subtract 14x from both sides.

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We could write this as x squared minus 14x plus 49 is

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equal to 0.

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Obviously, 14x minus 14x is 0.

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This quantity minus 14x is this quantity right there.

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Now we just have to think about what two numbers, when I

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take their product, I'm going to get 49, and when I take

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their sum, I'm going to get negative 14.

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So one, they have to be the same sign because this is a

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positive number right here.

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And they're both going to be negative because

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their sum is negative.

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And there's something interesting here.

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49 is a perfect square.

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Its factors are 1, 7, and 49.

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So maybe 7 will work, or even better, maybe

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negative 7 will work.

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And it does!

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Negative 7 times negative 7 is 49.

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And negative 7 plus negative 7 is negative 14.

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We have that pattern there, where we have 2 times a

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number, and then we have the number squared.

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This is a perfect square.

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This is equal to x minus 7, times x minus

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7, is equal to 0.

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Don't want to forget that.

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Or we could write this as x minus 7 squared is equal to 0.

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So this was a perfect square of a binomial.

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And if x minus 7 squared is equal to 0, take the square

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root of both sides.

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You'll get x minus 7 is equal to 0.

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I mean, you could say x minus 7 is 0 or x minus 7 is 0.

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But that'd be redundant.

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So we just get x minus 7 is 0.

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Add 7 to both sides, and you get x is equal to 7.

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Only one solution there.

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Let's do another one in pink.

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Let's say we have x squared minus 64 is equal to 0.

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Now this looks interesting right here.

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A bell might be ringing in your head on

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how to solve this.

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This has no x term, but we could think of it

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as having an x term.

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We could rewrite this as x squared plus 0x minus 64.

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So in this situation, we could say, OK, what two numbers,

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when I multiply them, equal 64, and when I

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add them equal 0?

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And when I take their product, I'm getting a

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negative number, right?

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This is a times b.

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It's a negative number.

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So that must mean that they have opposite signs.

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And when I add them, I get 0.

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That must mean that a plus minus b is equal to 0, or that

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a is equal to b, that we're dealing with the same number.

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We're essentially dealing with the same number, the negatives

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of each other.

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So what can it be?

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Well, if we're doing the same number and they're negatives

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of each other, 64 is exactly 8 squared.

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But it's negative 64, so maybe we're dealing with one

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negative 8, and we're dealing with one positive 8.

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And if we add those two together, we do

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indeed get to 0.

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So this will be x minus 8 times x plus 8.

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Now you don't always have to go through this

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process I did here.

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You might already remember that if I have a plus b times

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a minus b, that that's equal to a squared minus b squared.

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So if you see something that fits the pattern, a squared

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minus b squared, you could immediately say, oh, that's

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going to be a plus b-- a is x, b is 8-- times a minus b.

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Let's do a couple more of just general problems. I won't tell

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you what type these are going to be.

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Let me switch colors.

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It's getting monotonous.

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Let's say we have x squared minus 24x plus

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144 is equal to 0.

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Well, 144 is conspicuously 12 squared.

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And this is conspicuously 2 times negative 12.

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Or this is conspicuously negative 12 squared.

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So this is negative 12 times negative 12.

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This is negative 12 plus negative 12.

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So this expression can be rewritten as x minus 12 times

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x minus 12, or x minus 12 squared.

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We're going to set that equal to 0.

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This is going to be 0 when x minus 12 is equal to 0.

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You can say either of these could be equal to 0, but

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they're the same thing.

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Add 12 to both sides of that equation and you get x is

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equal to 12.

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And I just realized, this problem up here, I factored

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it, but I didn't actually solve the equation.

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So this has to be equal to 0.

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Let's take a step back to this equation up here.

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And the only way that this thing over here will be 0 is

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if either x minus 8 is equal to 0 or x plus

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8 is equal to 0.

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So add 8 to both sides of this.

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You get x could be equal to 8.

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Subtract 8 from both sides of this.

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You get x could also be equal to negative 8.

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Let's do one more.

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Just to really, really get the point drilled in your head.

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Let's do one more.

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Let's say we have 4x squared minus 25 is equal to 0.

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So you might already see the pattern.

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This is an a squared.

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This is a b squared.

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We have the pattern of a squared minus b squared,

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where, in this case, a would be equal to x, right?

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This is 2x squared.

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And b would be equal to 5.

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So if you have a squared minus b squared, this is going to be

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equal to a plus b times a minus b.

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In this situation, that means that 4x squared minus 25 is

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going to be 2x plus 5 times 2x minus 5.

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And of course, that will be equal to 0.

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And this will only be equal to 0 if either 2x plus 5 is equal

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to 0 or 2x minus 5 is equal to 0.

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And then we can solve each of these.

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Subtract 5 from both sides.

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You get 2x is equal to negative 5.

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Divide both sides by 2.

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You could get one solution is negative 5/2.

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Over here, add 5 to both sides.

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You get 2x is equal to positive 5.

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Divide both sides by 2.

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You get x could also be equal to positive 5/2.

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So both of these satisfy that equation up there.

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