Solving Quadratics by Completing the Square
October 6, 2026

[00:00:05] Right this now we're solving a quadratic
[00:00:11] Equation by factoring
[00:00:17] How do we so what's the first step we always do
[00:00:24] No
[00:00:29] Well before we do any of that, okay
[00:00:32] Oh
[00:00:34] We have to have a zero on one side, so what do I do subtract the 25? Yes
[00:00:49] Now what
[00:00:55] Oh
[00:01:00] Well in order to factor this what do we do first
[00:01:08] Multiply this out
[00:01:10] Oh
[00:01:19] And we got to distribute some stuff
[00:01:40] I
[00:01:42] Was that my 16
[00:01:47] Okay
[00:01:48] Yeah, we got to get zero to one on one of the two sides and then we got to put it in standard form
[00:01:55] standard form is
[00:01:57] some number of times x squared
[00:02:00] Plus the linear term plus the constant term
[00:02:04] We'll talk about that
[00:02:07] Now what? Now we have to factor.
[00:02:13] Got it, right
[00:02:18] What is it x plus 2
[00:02:28] And then what do we use
[00:02:33] Zero product property
[00:02:36] We have a product that equals zero
[00:02:39] Either x minus eight equals zero, or
[00:02:43] x plus 2 equals zero and now we no longer have a
[00:02:48] Quadratic equation we have two linear equations
[00:02:53] Thanks to the zero product property
[00:02:59] We got our two solutions
[00:03:09] This makes sense we did this last week the only thing we probably didn't do yeah was multiply something out
[00:03:22] That's solving by factoring: get everything to one side.
[00:03:27] Make sure zero is on the other side really
[00:03:30] Then we put in a standard form we factor it use zero product property and so on
[00:03:35] All right. Next question
[00:03:56] So
[00:04:00] By factoring we would get this to this side, right?
[00:04:05] And then how do we factor that
[00:04:08] Oh
[00:04:09] Difference of squares, okay. However, this is missing the linear term, right? So which property can we use?
[00:04:20] The SRP which stands for
[00:04:23] Square root principle.
[00:04:27] Square root.
[00:04:30] Principle.
[00:04:36] You equals plus or minus the square root of 25
[00:04:44] You equals plus or minus 5
[00:04:57] Which is two equations and we got our two solutions, right?
[00:05:08] Five and negative five, but I'm gonna go back for a sec
[00:05:17] When can we use the square root principle?
[00:05:24] When there's no what
[00:05:26] Linear term
[00:05:28] sometimes we call the coefficient of the
[00:05:30] Quadratic term we call it a the coefficient of the linear term
[00:05:36] We call it b, and the constant term we call c. Whenever b is zero, we can use the square root principle.
[00:05:43] What I failed to tell you was
[00:05:47] You actually equals X minus 3
[00:05:56] Okay
[00:05:59] Okay
[00:06:01] Okay, so we got X minus 3 squared equals 25
[00:06:13] Can we still use the square root principle
[00:06:19] We use it right here
[00:06:22] Here, x minus three equals plus or minus the square root of twenty-five.
[00:06:36] X minus 3 is plus or minus 5
[00:06:42] X is 3 plus or minus 5
[00:06:46] X equals 3 plus 5 or X equals 3 minus 5
[00:06:53] Okay, that's a great question why did we not do that right here one is got told you just solved by factoring
[00:07:02] Two is the whole point is
[00:07:05] To show us even though this is not X squared equals the number
[00:07:10] This is something
[00:07:12] squared equals a number
[00:07:16] Okay, let's call this
[00:07:19] You for a second we got u squared equals 25 we could still use the square root principle
[00:07:28] Okay, whatever is squared
[00:07:35] Whatever is squared this case is not just X it's X minus 3 we simply go X minus 3
[00:07:42] Equals plus or minus the square root of the other side and that is using the square root principle
[00:07:51] Okay
[00:07:54] Go with me for a second
[00:07:57] Okay
[00:07:58] It saves us a lot of time. We don't have to multiply out
[00:08:02] and bring everything over and then
[00:08:05] factor again
[00:08:08] Okay, so let me do another example
[00:08:11] We
[00:08:11] Have something squared equals a number
[00:08:18] Can we use the square root principle
[00:08:22] Yeah, whatever is squared let me call it you for a second
[00:08:27] X plus five equals plus or minus the square root of nine.
[00:08:35] X plus five equals plus or minus three.
[00:08:41] X equals five plus or minus three.
[00:08:46] Which is X two different equations
[00:08:56] We get 8
[00:09:00] 2
[00:09:04] Negative five! Yes, thank you.
[00:09:13] Negative 2 and X equals negative 8 we got our two solutions
[00:09:20] Oops
[00:09:31] Okay, so we could use the square root principle
[00:09:37] When anything squared equals a number
[00:09:41] whatever squared
[00:09:44] Equals plus and minus the square root of the other side, which is two separate equations, and then we we go from there
[00:09:51] We're gonna solve this by well, we're sorry this can easily be solved by factor, right?
[00:10:00] Okay, what's this factor to
[00:10:08] Actually, let me do a different example
[00:10:16] This equals x plus five times x plus one. Then what do we do?
[00:10:27] Zero product property X plus 5
[00:10:32] X plus 1
[00:10:43] Actually, let me do a different problem
[00:10:55] Now let's do with this sorry, okay, let's take the same equation
[00:11:10] I
[00:11:13] Put this on this side
[00:11:27] I'm going to add something to both sides
[00:11:33] Okay, so when this factors it's a perfect square
[00:11:51] So I'm gonna add some number
[00:11:56] So it will factor to the same exact two binomials
[00:12:03] This gives a perfect square. What do you think we should add?
[00:12:11] Okay, she's saying add 9
[00:12:22] If you add 9 we'll talk about a second how we do that what is this factor to
[00:12:29] X plus 3 X plus 3
[00:12:32] X plus 3
[00:12:34] Okay, we'll come back this minute. Let's just finish this
[00:12:47] And once we're here
[00:12:50] What can we use now? What's something squared equals number square root principle?
[00:12:55] We get X plus 3 is plus or minus the square root of 4
[00:13:07] Which is two equations
[00:13:21] There are two same two solutions we solved by factoring earlier ago
[00:13:29] Once we write our two binomials
[00:13:34] These two
[00:13:37] Have to multiply to be this
[00:13:42] And then
[00:13:44] These two 3x and 3x have to add to be 6x
[00:13:52] Okay
[00:13:55] So we literally just take this B value divide by 2 and that's going to be the number here in here
[00:14:06] Okay, when we factor it which is B over 2
[00:14:12] And then
[00:14:17] To figure out what number we add we just take B over
[00:14:22] 2 and we square it and that's the number we add so 6 over 2
[00:14:28] Squared this 3 squared is 9 and that's what we add to both sides of the equation
[00:14:36] Okay, this is called solving
[00:14:44] By completing
[00:14:49] The square
[00:14:53] Okay, why is it called solving by completing this square because once we're here
[00:15:02] We have to make we have to add something to make it a perfect square so we need to complete the square and
[00:15:11] That's we always add B over 2 squared
[00:15:15] And we got this we now completed this square. It's a perfect square when it factors
[00:15:19] It's gonna be something square a binomial squared
[00:15:25] Okay, let's try another one
[00:15:28] Here's a quadratic
[00:15:34] Polynomial that does not factor
[00:15:39] Okay
[00:15:42] Well, it actually does we'll talk about that in a second
[00:15:46] But the only options we know of our what?
[00:15:50] Negative 5 and 1 or
[00:15:53] five and negative one, and neither pair adds to eight.
[00:15:58] We agree. Yes. Okay
[00:16:01] So this cannot be solved by factoring
[00:16:06] The only other way we know of is by completing the square so the first step
[00:16:16] We get the constant term on the right the quadratic and the linear on the left
[00:16:26] And now we got to add something
[00:16:32] So becomes a perfect square
[00:16:37] You got to complete the square, so what am I gonna add?
[00:16:42] These two numbers have to add to eight.
[00:16:46] So which is
[00:16:49] 8 over 2 is
[00:16:52] 4 so it's plus 4 plus 4
[00:16:56] And then to figure out they have to multiply to be this number so we square
[00:17:02] 16
[00:17:04] We add sixteen to both sides.
[00:17:21] And now we have something squared equals a number. What can we use?
[00:17:26] Square root principle
[00:17:31] X plus four equals plus or minus the square root of twenty-one.
[00:17:36] Does the square root of twenty-one simplify? No.
[00:17:42] Okay
[00:17:50] Okay, and those are the two solutions
[00:17:55] It's negative 4
[00:17:56] plus the square root of twenty-one,
[00:17:59] and negative four minus the square root of twenty-one.
[00:18:03] Both solutions are irrational numbers
[00:18:10] But that's it
[00:18:12] So we were able to solve this
[00:18:16] Even though it could not be solved by factoring
[00:18:22] Okay, and let me show you something
[00:18:30] This
[00:18:31] Actually does factor
[00:18:36] This would be x and x, right? It's simply
[00:18:45] For
[00:18:48] four plus the square root of twenty-one, and four minus the square root of twenty-one.
[00:18:54] So it's fine
[00:18:59] Okay, and there's no way we were gonna guess that right
[00:19:03] The only we guess is integer values
[00:19:07] Okay, and these these two numbers have the same property
[00:19:12] they multiply to be negative 5 and
[00:19:17] They add to be 8
[00:19:20] Okay, they're just not nice whole numbers or integers I should say
[00:19:27] These add to eight: four plus the square root of twenty-one plus four minus the square root of twenty-one.
[00:19:34] That's 0 we get 8
[00:19:37] And do they multiply to be negative 5?
[00:19:41] They do let's try it
[00:19:53] 16
[00:19:56] Minus four times the square root of twenty-one, plus four times the square root of twenty-one,
[00:20:04] Minus
[00:20:05] 21
[00:20:07] These terms cancel, and sixteen minus twenty-one gives negative five.
[00:20:11] It's kind of cool. That's pretty cool. It is pretty cool
[00:20:20] Okay, let's try that one and then we'll get to the more difficult
[00:20:24] Again there's no way that's
[00:20:27] Solve this by factoring. So we're gonna complete the square first step
[00:20:36] We gotta get the constant over here and then we complete the square
[00:20:52] We take this number divide by 2 and square
[00:21:07] We add twenty-five to both sides.
[00:21:19] And this definitely factors to
[00:21:23] Half of that number would be over 2 so that's plus 5 plus 5
[00:21:37] And now what square root principle
[00:21:45] X plus five equals plus or minus the square root of twenty-eight.
[00:22:11] Does the square root of twenty-eight simplify? I should have done that first.
[00:22:21] Oh
[00:22:22] There we go, we got our two solutions
[00:22:33] We can now solve quadratic equations when they cannot be factored over the integers,
[00:22:38] or at least are not obviously factorable.
[00:22:44] Let's try that one
[00:22:46] Okay, what's the first thing we do?
[00:22:54] We got to get the constant on the right
[00:23:04] Okay, and we have a little problem on this one
[00:23:10] We want to get x plus or minus some number
[00:23:18] We want just an x here
[00:23:22] We could use square root of two times square root of two,
[00:23:27] Because then that does multiply to be 2x squared
[00:23:31] But it's way too complicated
[00:23:35] Okay, so
[00:23:37] Whenever the leading coefficients not one
[00:23:39] We have to divide both sides by the leading coefficient. We need to change the leading coefficient to one
[00:23:47] So we divide by 2 divided by 2
[00:23:50] We get x squared minus 3 over 2x
[00:23:55] 5 over 2
[00:24:10] And now we complete the square
[00:24:17] And now we're gonna deal with some fractions and that's okay, this is my B value
[00:24:24] Negative three over two. Take b over two, squared.
[00:24:32] Which is negative 3 over 4 squared which is
[00:24:44] 9 16th
[00:24:46] So we add 9 16th
[00:25:01] It's that factors to x plus or minus
[00:25:06] Minus minus what?
[00:25:09] 3 over 4 it's the number of before we square it or it's also the square of that but then you got to take care of
[00:25:18] the sign
[00:25:25] Okay, and then we got to get a common denominator over here
[00:25:35] That's 40 plus 9 49 over 16
[00:25:41] That right, yes
[00:25:52] Now what
[00:25:53] Something squared equals a number. Use the square root principle: x minus three fourths
[00:26:03] equals plus or minus the square root of 49 16th
[00:26:16] Was that 7 over 4?
[00:26:50] Which is 10 fourths or negative 4 fourths?
[00:27:04] which is five halves. Yes.
[00:27:20] This happened to be a perfect square, so we ended up with no square roots. When that number is not a perfect square,
[00:27:27] You're gonna end up with square roots in your solution
[00:27:37] We're gonna do it
[00:27:42] You try one more
[00:27:43] Actually, what's the first step? Okay, that's good
[00:27:57] Yeah, man, what's the next step? Okay, you guys are paying attention I like this
[00:28:13] Divide what by two
[00:28:17] Yeah, we sometimes call that the B value
[00:28:39] right, we got to add it to complete the square
[00:28:57] Oh
[00:28:59] Six goes into 144 24. Yes
[00:29:13] What's five times 24 120 plus 49 is
[00:29:27] One hundred sixty-nine.
[00:29:38] Common say we can't do that because uh, this is 13
[00:29:43] Times 13 and this is 12 times 12, but those don't simplify
[00:29:50] But you might be saying something different
[00:29:55] We use the square root principle.
[00:29:58] X minus seven twelfths equals plus or minus the square root of one hundred sixty-nine over one hundred forty-four. You were ahead of me, I think.
[00:30:10] Okay, clear it now what
[00:30:32] I
[00:30:33] Like that
[00:30:39] Negative six twelves
[00:30:49] Twenty twelfths which is
[00:30:56] Five thirds
[00:31:17] Oops
[00:31:18] Not an ordered pair: the solution set.
[00:31:29] That was luck it does tell us whenever you get fractions here well
[00:31:40] What was the original equation
[00:31:46] Okay
[00:32:05] This
[00:32:06] This does not factor to this why not
[00:32:10] Because
[00:32:12] Because
[00:32:15] The leading term would not be six x squared, correct?
[00:32:20] However, it's right to X times X is not six X square
[00:32:24] However, all you got to do is stick a six out here now. It does factor into that
[00:32:30] And if you change this to two times three
[00:32:39] Like
[00:32:46] You distributed the
[00:32:49] Two to this guy you get two X plus one and
[00:32:57] Three into this factor gives three
[00:33:00] X
[00:33:01] So we actually could solve this by factoring Wow
[00:33:09] But
[00:33:11] That's kind of crazy
[00:33:13] Okay, all right two numbers that multiply to be negative 30 add to be negative 7 or what?
[00:33:32] So we could have solved this by factoring
[00:33:35] changes to 6x squared
[00:33:37] Minus 10x plus 3x minus 5 and a factor by grouping you would have ended up with this
[00:33:44] Wow, okay
[00:33:47] But we didn't need to solve by completing square any
[00:33:54] Any quadratic equation can be solved by completing the square.
[00:34:00] Once in a while and give you an example
[00:34:15] Let's
[00:34:15] Try this one real quick
[00:34:19] We'll be the first step
[00:34:23] Subtract five on both sides. That's right.
[00:34:29] That's right: b over two, squared. Two over two,
[00:34:34] squared
[00:34:35] 1 squared 1 so we add 1 to both sides
[00:34:51] Then what do we use
[00:34:56] Square root principle: x plus one equals plus or minus the square root of negative four.
[00:35:07] And what is the square root of negative 4
[00:35:13] It's not a real number, okay
[00:35:19] Not real
[00:35:21] There are no real numbers whose square is negative four.
[00:35:29] So what does that tell us about the original?
[00:35:32] equation
[00:35:34] There are no real solutions.
[00:35:36] We often say real because we're gonna learn something else later
[00:35:42] Does it have a solution set yes, yes it is
[00:35:47] The empty set
[00:35:50] Okay
[00:35:51] So if once you use the square root principle and the right hand side is a negative you're done
[00:35:59] No real solutions
[00:36:01] Right and to think about it. There's nothing that you could square
[00:36:05] A real square is nonnegative, so it cannot equal this negative number.