Solving Quadratics by Factoring and Square Roots
October 1, 2026

[00:00:05] What are we getting an equation here?
[00:00:13] direction or solve with me. Solve
[00:00:15] find the solution set.
[00:00:17] That's great.
[00:00:19] Set of all solutions.
[00:00:22] What do we call this type of equation?
[00:00:29] This is called a linear equation.
[00:00:32] Why is this called a linear equation?
[00:00:35] The variable is only to the first power.
[00:00:41] I'll just tell you.
[00:00:43] This is what type of polynomial?
[00:00:45] Linear polynomial.
[00:00:48] A linear polynomial. Yes, it happens to be binomial.
[00:00:52] So this is a linear polynomial.
[00:00:57] And what is this?
[00:01:00] Constant.
[00:01:02] Polynomial.
[00:01:04] Definitely a monomial.
[00:01:06] Okay.
[00:01:09] The linear is the larger degree.
[00:01:12] Since it's the larger degree,
[00:01:15] the entire thing is called linear. Sorry, a linear equation.
[00:01:19] A linear equation.
[00:01:23] You could call it a linear polynomial equation.
[00:01:26] We simplify it. Linear equation.
[00:01:29] And we do a bunch of steps.
[00:01:32] What's the first step?
[00:01:34] Subtract seven.
[00:01:38] We get 3x
[00:01:42] equals two.
[00:01:43] And then we divide by three.
[00:01:47] Divide by three.
[00:01:49] Just two thirds.
[00:01:51] And then the solution is
[00:01:54] two thirds.
[00:01:57] And the solution set is
[00:02:00] the set containing two thirds.
[00:02:05] This process we went through,
[00:02:07] we learned in, I don't know, seventh grade?
[00:02:09] Is often known as isolating the variable.
[00:02:23] Once we got the variable all by itself,
[00:02:26] it's obvious what the solution is.
[00:02:29] It's the number on the other side.
[00:02:31] Because when you replace it for the x,
[00:02:34] makes the equation true.
[00:02:37] Okay.
[00:02:41] Next.
[00:03:03] What do we call this side of the equation?
[00:03:07] It's a quadratic equation.
[00:03:10] The left side is a quadratic polynomial,
[00:03:12] the right side is a constant polynomial,
[00:03:14] the larger the two degrees is two,
[00:03:16] the quadratic.
[00:03:17] We call the entire thing a quadratic equation.
[00:03:23] This is what we're talking about solving quadratic equations today.
[00:03:31] So, what are we going to do?
[00:03:47] We're going to isolate the variable.
[00:03:51] We want to get x equals how we got a problem.
[00:03:59] What's the problem?
[00:04:04] There's an x squared and there's an x.
[00:04:06] There's no way to combine those into one, right?
[00:04:09] Because they're different terms.
[00:04:11] You got an idea?
[00:04:14] Okay, we could factor out the x.
[00:04:18] Let's do that for a second.
[00:04:23] The x plus two.
[00:04:35] But there's no way to still...
[00:04:37] We still have two different x's.
[00:04:44] So now what?
[00:04:45] Subtract 15 and make it a quadratic trinomial.
[00:04:49] We want to factor them out.
[00:04:51] Okay, why would we do that?
[00:04:54] Because it will equal zero.
[00:05:04] Okay, how about this?
[00:05:09] Well, we're looking for a number and a number plus two
[00:05:13] more than that same number.
[00:05:16] It equals 15.
[00:05:21] We could start guessing and checking,
[00:05:23] but that's a terrible way.
[00:05:25] Okay, we have a very, very important property.
[00:05:30] Bet you even know what it is.
[00:05:35] It's called the zero product property.
[00:05:38] Okay.
[00:05:48] If a times b equals zero...
[00:05:57] If two numbers multiplied to be zero...
[00:06:02] What has to be true?
[00:06:06] Either a is zero...
[00:06:11] Or b equals zero.
[00:06:14] Or they both could be zero, correct?
[00:06:18] Okay, so that's still the zero product property.
[00:06:25] Why is it called that?
[00:06:27] Then why is it called the zero product property?
[00:06:30] Because it equals zero.
[00:06:31] Yeah, because we have a product that equals zero.
[00:06:35] This product equals zero.
[00:06:37] It's called the zero product property.
[00:06:39] Sometimes it's called the zero factor property.
[00:06:44] We have two factors that equal zero,
[00:06:48] but we're just going to call it the zero product property.
[00:06:52] Okay, so let's go back here.
[00:06:55] We have a product that equals 15.
[00:07:02] Okay.
[00:07:08] There's no...
[00:07:09] Well, let me give you this.
[00:07:11] What are some numbers that multiply to 15?
[00:07:15] Three and five.
[00:07:17] What else?
[00:07:18] 15 and one.
[00:07:20] 15 and one. What else?
[00:07:23] Negative three, negative five.
[00:07:24] Negative three, negative five.
[00:07:27] What else?
[00:07:27] Negative three, negative five.
[00:07:30] What else?
[00:07:35] How about 30 and one half?
[00:07:40] Negative 30, negative one half.
[00:07:41] How about 60 and a fourth?
[00:07:45] How many numbers multiply to be 15?
[00:07:51] More than a lot.
[00:07:54] An infinite.
[00:07:54] Okay.
[00:07:56] So we don't have a 15 product property,
[00:08:04] but we do have a zero product property.
[00:08:07] If we have zero, we've got only two options.
[00:08:10] Either the first number is zero or the second number is zero.
[00:08:16] So we are going to start with this.
[00:08:19] So we're going to go backwards.
[00:08:21] We have to have zero on one side.
[00:08:22] So we start by subtracting 15.
[00:08:28] Yes, people did say this before.
[00:08:33] And now we need to change this to a product.
[00:08:40] And that process is called...
[00:08:44] Factoring.
[00:08:44] Factoring.
[00:08:44] Thank you.
[00:08:46] We want to change our sum to a product that's called factoring,
[00:08:50] to write as a product.
[00:08:51] And so we're going to factor this.
[00:08:56] What's this factor to?
[00:09:01] Five and negative three.
[00:09:03] Five and negative three.
[00:09:04] Oops.
[00:09:09] Okay.
[00:09:12] Now we have a product that equals zero.
[00:09:15] Oh, so it's...
[00:09:17] By the zero product property,
[00:09:28] this number, x minus three, equals zero.
[00:09:33] Or this number, x plus five, equals zero.
[00:09:42] Now we don't have a quadratic equation.
[00:09:46] We have two what's that?
[00:09:48] Linear.
[00:09:49] Linear equations.
[00:09:51] And now all we have to do is isolate the variable.
[00:09:59] Add three.
[00:10:04] And the solution is three.
[00:10:08] And for x plus five, subtract five from both sides.
[00:10:11] Solution is negative five.
[00:10:15] Okay.
[00:10:17] And there it is.
[00:10:20] So once we turned it into two linear equations,
[00:10:23] we're able to just isolate the variable
[00:10:26] and get our solution set.
[00:10:28] And our solution set is negative five and three.
[00:10:33] Okay.
[00:10:35] So this is how we're going to solve quadratic equations.
[00:10:40] We need one side equals zero.
[00:10:43] Then we're going to factor.
[00:10:46] Okay.
[00:10:47] And then we use the zero product property,
[00:10:50] write two separate linear equations.
[00:10:54] And then we get our two solutions, if there are two solutions.
[00:11:00] Is that cool?
[00:11:02] Okay.
[00:11:02] We solved our first quadratic equation.
[00:11:06] This was the major point of why we spent so much time factoring.
[00:11:10] Okay.
[00:11:11] It allows us to solve quadratic equations,
[00:11:13] which before was impossible.
[00:11:16] You didn't know how to factor,
[00:11:17] and we didn't even know what factoring was before.
[00:11:21] Let's try another one.
[00:11:42] Okay.
[00:12:04] So what's the first step?
[00:12:09] We must have zero on one side.
[00:12:11] We don't care what side.
[00:12:13] We must have zero on one side.
[00:12:15] So we subtract seven X.
[00:12:18] And what order are we in writing?
[00:12:22] Descending order.
[00:12:30] And once we do that, what do we do?
[00:12:33] We need a factor.
[00:12:40] And what's this factor, two?
[00:12:44] Two numbers that multiply to be negative 30,
[00:12:46] that add to be negative seven.
[00:12:49] Negative 10, positive three.
[00:12:54] Okay.
[00:12:54] And then once we have a fact or a product that equals zero,
[00:12:58] what do we use?
[00:13:00] Zero product.
[00:13:03] Zero product.
[00:13:10] Either X minus 10.
[00:13:14] This number equals zero.
[00:13:17] Or X plus three.
[00:13:21] And now we got two linear equations.
[00:13:24] And we solve those.
[00:13:47] And we got our two solutions.
[00:13:54] Okay.
[00:13:57] Quadratics can have at most two equations,
[00:14:01] sorry, two solutions.
[00:14:03] It could have one solution.
[00:14:05] We'll give an example of that in a minute.
[00:14:08] Or it can have no solutions.
[00:14:11] And we'll give an example of that.
[00:14:12] But most of the time it's going to have two solutions.
[00:14:16] Let's try another one.
[00:14:39] Step one.
[00:14:42] We're not setting it to zero.
[00:14:44] We need to get one side to be zero.
[00:14:48] And we do that by?
[00:14:50] Subtracting 100.
[00:14:58] And now we need to factor that quadratic.
[00:15:07] Factors of negative 300 that add to be negative 40.
[00:15:29] You know what, I screwed up.
[00:15:32] Let's make this a negative 100.
[00:15:34] Sorry, I messed up the problem.
[00:15:39] So this becomes positive.
[00:15:43] So sorry, put that in there.
[00:15:46] Otherwise it would not be able to be factored.
[00:15:49] Now we're looking for factors of 300 that add to be negative 40: negative 10 and negative 30.
[00:16:02] Negative 10, thank you.
[00:16:05] Negative 30.
[00:16:09] Minus 10x minus 30x plus 100.
[00:16:18] And factor it by grouping.
[00:16:50] Okay, once we factor, we have a factor that equals, or a product that equals zero.
[00:16:59] What's next?
[00:17:03] So by the zero product property.
[00:17:12] And when you're solving this for homework, I want you to write that in every time you use it.
[00:17:19] You can abbreviate it if you want.
[00:17:21] What should we abbreviate it as?
[00:17:24] ZPP.
[00:17:26] Either x minus 10 equals zero, or 3x minus 10 equals zero.
[00:17:44] And we have our two solutions, 10 thirds and 10.
[00:17:50] How can we check to see if our solutions are actual solutions?
[00:17:56] We can plug it back into the original and?
[00:18:02] What does it mean to be a solution?
[00:18:06] A number that, when replaced for the variable, makes the equation true.
[00:18:10] If you plug it in, it makes the equation true.
[00:18:12] If you factor right, it's going to be the solutions.
[00:18:17] Okay.
[00:18:35] This is a binomial.
[00:18:37] What do I write?
[00:18:38] Binomial.
[00:18:40] The only one we know how to factor is what?
[00:18:42] Difference of squares.
[00:18:46] Is this the difference of squares?
[00:18:50] So if factors two, x minus seven, x plus seven.
[00:19:05] And now what?
[00:19:09] Zero.
[00:19:10] Zero.
[00:19:14] X minus seven equals zero, x plus seven equals zero.
[00:19:19] X equals seven, x equals negative seven.
[00:19:29] What?
[00:19:33] No.
[00:19:44] What?
[00:19:52] This is not the difference of squares.
[00:19:54] What is this?
[00:19:55] Sum of squares.
[00:19:58] Do sum of squares factor?
[00:20:01] No.
[00:20:05] Doesn't factor.
[00:20:06] That is not a factor.
[00:20:13] What does that mean?
[00:20:18] There's no solutions.
[00:20:21] There is a solution set.
[00:20:22] What is it?
[00:20:23] Empty set.
[00:20:24] The empty set.
[00:20:25] Either that or that.
[00:20:27] Okay.
[00:20:28] Just because a polynomial doesn't factor doesn't mean there's no solutions.
[00:20:34] We'll get to that later.
[00:20:36] But this one definitely does not factor.
[00:20:39] There's no solutions.
[00:20:43] You could also think this way.
[00:20:46] X squared equals negative twenty-five.
[00:20:51] There's no way or something squared which will always be positive to equal a negative number.
[00:20:56] There's definitely no solutions here.
[00:21:18] Is this the difference of squares?
[00:21:22] Yeah.
[00:21:23] It is.
[00:21:30] Two x, squared, is four x squared.
[00:21:34] What squared is 5?
[00:21:35] Square root of 5.
[00:21:37] Square root of 5.
[00:21:51] And now what?
[00:21:55] ZPP.
[00:21:57] We have a product that equals zero.
[00:21:59] We set each factor equal to zero.
[00:22:30] And we got negative the square root of 5 over 2.
[00:22:36] And the square root of 5 over 2.
[00:22:42] Sometimes when we have the same number but just opposite signs, well it's not the same number,
[00:22:46] but opposite numbers.
[00:22:49] We sometimes write it like this.
[00:22:51] Plus and minus the square root of 5 over 2.
[00:22:58] Like we could have done that for this one.
[00:23:02] We could have wrote plus and minus 7 which is two different numbers.
[00:23:22] Yeah, I think we got time.
[00:23:32] We had this equation before.
[00:23:34] Yes.
[00:23:37] Okay.
[00:23:44] I agree this is the equivalent equation.
[00:23:48] Yes.
[00:23:55] Can I just take the square root of both sides?
[00:24:05] Okay.
[00:24:07] Possibly.
[00:24:10] But then we only get one solution.
[00:24:21] So just taking the square root of both sides, you only get one solution.
[00:24:27] Okay.
[00:24:36] However, we know this can be factored, right?
[00:24:41] This can be factored to X minus 7 times X plus 7.
[00:24:49] And then by the ZPP, we get X minus 7 equals 0 and X plus 7 equals 0.
[00:25:03] And we got our two solutions, negative 7 and 7.
[00:25:07] Okay.
[00:25:13] So if X squared equals any number, let's call it A.
[00:25:20] A has to be greater than or equal to 0.
[00:25:27] X squared minus A equals 0 is equivalent.
[00:25:41] Is this the difference of squares?
[00:25:44] Yeah.
[00:25:46] X squared minus the square root of a squared factors to x minus the square root of a, times x plus the square root of a.
[00:26:01] And by the zero product property.
[00:26:08] X minus the square root of a equals zero, so x equals the square root of a.
[00:26:19] And X equals negative square root of A.
[00:26:22] And we get negative the square root of a.
[00:26:34] Which could be written as plus or minus the square root of A.
[00:26:40] And that number, number A does not matter.
[00:26:44] As long as a positive number.
[00:26:47] Why is the number of the negative number?
[00:26:51] What'd you say?
[00:26:55] This works always, except for when A is a negative number.
[00:27:00] It does.
[00:27:02] The square root of negative numbers don't exist.
[00:27:07] There's no real numbers.
[00:27:10] Okay.
[00:27:10] So if any time we have X squared equals a positive number, we know the two solutions are X equals the square root of that number.
[00:27:30] And X equals negative the square root of that number.
[00:27:33] We just proved it.
[00:27:35] Okay.
[00:27:36] We got these two equations.
[00:27:40] Okay.
[00:27:41] And we often write it like this.
[00:27:42] X equals plus or minus the square root of that number.
[00:27:50] Okay.
[00:27:51] So if this is true, then this is true.
[00:27:55] Let's write that down.
[00:27:58] If X squared equals a number, as long as positive, then X equals plus and minus the square root of A.
[00:28:11] Which is two equations.
[00:28:12] X equals the square root of A.
[00:28:15] And X equals the negative square root of A.
[00:28:20] Okay.
[00:28:21] This is called the square root principle.
[00:28:36] And it's going to allow us to solve quadratic equations without factoring.
[00:28:44] Because we just proved it always factors.
[00:28:47] So for example, if we have X squared equals 25.
[00:28:57] It looks like you're taking the square root of both sides, but we're not.
[00:29:01] It's by the square root principle.
[00:29:15] That X will equal plus or minus the square root of 25.
[00:29:20] Okay.
[00:29:21] We're not taking the square root of both sides.
[00:29:25] We're using the square root principle.
[00:29:27] Anytime something squared equals a number, whatever is squared equals plus or minus the square root of that number.
[00:29:33] Which is two separate equations.
[00:29:36] Okay.
[00:29:38] Plus or minus five.
[00:29:40] And that gives our two solutions.
[00:29:46] Okay.
[00:29:49] So, anytime you have X squared, let's just do another one.
[00:30:01] Equals 50.
[00:30:06] We're not taking the square root of both sides.
[00:30:09] We're going to use the what principle?
[00:30:12] Square root principle.
[00:30:14] Square root principle.
[00:30:20] Then X will equal plus or minus the square root of 50.
[00:30:27] And what does that simplify to?
[00:30:31] Five square root of two.
[00:30:34] Five square root of two.
[00:30:37] We have our two solutions.
[00:30:41] Okay.
[00:30:42] All the square root principle is saving time where we're not bringing this to the other side.
[00:30:48] And we're not factoring.
[00:30:56] And we use this as ZPP.
[00:31:12] Okay.
[00:31:22] We got our two solutions.
[00:31:25] Our two equations.
[00:31:27] Okay.
[00:31:28] So, because we know it's going to be the difference of squares, there's no linear term.
[00:31:35] Okay.
[00:31:36] It will factor and we'll end up with these two equations.
[00:31:40] Okay.
[00:31:41] And we write this all in one equation.
[00:31:44] Right.
[00:31:44] We could.
[00:31:48] Okay.
[00:31:48] So, that is the square root principle.
[00:31:50] We can only do it when we have something squared equals some positive number.
[00:31:58] Okay.
[00:31:58] And you go right to this equation.
[00:32:01] Does this make sense?
[00:32:03] Yes.
[00:32:04] This is going to be very important for the next thing we learn later.
[00:32:12] Okay.
[00:32:13] I think we could stop here.
[00:32:16] Most of the problems, we're not going to be able to use the square root principle.
[00:32:20] We just have to factor and use what property?
[00:32:26] Zero product property.
[00:32:28] And you need to write in there where you use it.
[00:32:31] You write two separate equations and then solve.