3 ways to "complete the square"

[@blackpenredpen]

Have you impressed your teachers (or students) yet???

[@Ashmit_1729]

Yes by factoring 2

[@davidbrisbane7206]

Yes indeed. When I showed them how to use Wolfram Alpha to solve quatratic equations 😁.

[@أميرالبلابل]

Yeah, when I correct my teacher hahahah

[@invaderzim6842]

Hi ,blackpenredpen!I really enjoy your content. I’ll very much appreciate if u can solve me this problem Find x for:2^((sin(x))^2014)-2^((cos(x))^2014)=(cos(2x))^2013 I know it is a tough problem ,but I really want to know the way to solve this kind of problem 😅

[@sbusisoquintin8004]

Not yet we are waiting for them😅💯 Support from South Africa🇿🇦

[@shaurya1616]

I think his immense knowledge comes from the pokeball.

[@yensteel]

Agreed

[@mama5402]

I guess that's why he looks like a pokemon xd...

[@arjunsahu2543]

😂🤣🤣

[@williamsmamanimamani846]

Estoy totalmente de acuerdo, mi estimado

[@xxmysticexpertxx113]

Pikachu tellin him the answers

[@justacityboy4426]

Math is basically magic, but without all the magic

[@Exahax101]

It's logic not magic!

[@SofronPolitis]

That's why it's called mathemagics

[@Felixrizzlord]

@@Exahax101 yes

[@E12345E]

Given: Math is basically magic, but without all the magic Prove: Magic doesn’t exist math = magic Given math - magic = magic Given magic - magic = magic substitution POE 0 = magic Simplify this proves that magic does not exist

[@johnbiluke8406]

@@E12345E (YEAH, YOU IN THE CROWD, CAN YOU SPOT THE MISTAKE?)

[@markdougherty8203]

Method 3 is just completing the square from the other side. By symmetry it works the same. It's a cool trick though, because most people don't expect it. I have surprised people with it before. Obviously it's more convenient if the units term is a perfect square.

[@SyberMath]

You have a way with Complex Numbers! Nice work!!! 🤩

[@blackpenredpen]

Thanks! 😃

[@SyberMath]

@@blackpenredpen My pleasure! 😊

[@vivekbhutada3049]

Hey syber! Big fan here!!

[@erikmensinga]

@marine dtadtfeld bruh no why

[@AliKhanMaths]

I love the second way the most - looks so clean! You've really inspired me to share my maths tricks too!

[@idrisShiningTimes]

I loved the first method honestly, since I'm basically used to solving fractions, and it gives out the answer in simplest form most of the time.

[@shuvo1562]

Type 3 seems new to me!Every time I learn something new from your videos.

[@beaming_sparkling_trash261]

I'm in love w the third method

[@Firefly256]

Irrational Denominator to rational = rationalize Imaginary denominator to real = realize

[@blackpenredpen]

Try this cubic x^3-3x^2-3x-1=0 Hint: it's kinda similar to the 3rd way in this video. Solution: kzbin.info/www/bejne/gKuQi2Ogm9hljJY

[@crispyclips6268]

Bro can you please post some videos on calculus (majorly integral calculus) and functional equations too As I am preparing for jee advanced and the questions need practice for developing that approach. How to contact you ?

[@crispyclips6268]

And more questions like this one kzbin.info/www/bejne/aandk2B_pr53fM0

[@4ltrz555]

@@crispyclips6268 he has made a lot of integration speedruns and long runs. You can check it out in his playlists

[@crispyclips6268]

OK thanks

[@RazorM97]

At 5:45 i thought you were going to say, "let's delete that two and it will work out nicely". Man I was so wrong XD

[@blackpenredpen]

Do you have a different way to complete the square for this equation?

[@ryancantpvp]

No because the video is still premiering lol!

[@purushothaman6698]

No

[@mathevengers1131]

Fourth method: Use quadratic formula Fifth way: Tell πkachu to use thunderbolt

[@beastgamer4932]

how do you use quadratic formula to complete the square

[@davidbrisbane7206]

We can set x = 1/y, as x = 0 is not a solution. So, 2x² + 6x + 9 = 0 is transformed into 9y² + 6y + 2 = 0 So, 36y² + 24y + 8 = 0 So, (6y + 2)² = -4 So, 6y + 2 = ±2i So, 3y = -1 ± i So, y = (-1 ± i)/3 So, x = 3/(-1 ± i) So x = (-3 ± 3i)/2

[@MathAdam]

Next video: 3 Ways to Complete the Pikachu

[@anshumanagrawal346]

Yeesss

[@paulg444]

This guy is the man!!!.. Love his energy!

[@prepa-maths]

Hello from France !!! Excellent channel !!

[@tambuwalmathsclass]

I must try this in with my students

[@juancarlosquispemamani7238]

Genial!!! yo soy de Bolivia voy en la carrera de Ingeniería y voy aprendiendo mucho de ti... muchas gracias por compartir tus conocimientos!!!! sigue adelante!!! exitos!!!

[@SandeepKumar-dw3sj]

Wow you have legend way to solve it

[@patrickpablo217]

Nice! All three were great :) I'll add another twist that can fit with any of the three. Recently I switched from "completing the square" to "completing the difference of squares" and I like it a lot more. I'll give a basic example: x^2+6x+8=0 proceed as usual but *don't* move anything to the RHS: x^2 + 6x + 3^2 + -(3)^2 + 8 = 0 (x + 3)^2 - ( 3^2 - 8 ) = 0 that second term isn't a square yet but we can write it as one easily: (x + 3)^2 - (√[ 3^2 - 8 ])^2 = 0 Now we have a difference of squares, but let's simplify first: (x + 3)^2 - (√[ 9 - 8 ])^2 = 0 (x + 3)^2 - (√[ 1 ])^2 = 0 (x + 3)^2 - (1)^2 = 0 now difference of squares takes care of the ± automatically and gives (x+3+1)*(x+3-1)=0 (x+4)*(x+2)=0 so x={2,4} I really like this method since you never have to deal with moving things from LHS to RHS or back, and it's more like factoring, and it avoids the ± issue completely. It also has direct relation to the parabola associated with the quadratic: the first of the squares gives the line of symmetry (the x-coordinate of the vertex), and the second one provides the distance to the solutions and the y-coordinate of the vertex. And it saves steps, which is nice.

[@patrickpablo217]

for this equation 2*x^2 + 6x + 9=0: (doing more arithmetic than necessary, just to not get people lost) 2*x^2+6x+(3/√2)^2 -(3/√2)^2 + 9 =0 (√2*x+(3/√2))^2 - ((3/√2)^2 - 9) = 0 (√2*x+(3/√2))^2 - (9/2 - 9) = 0 (√2*x+(3/√2))^2 - (-9/2) = 0 (√2*x+(3/√2))^2 - (i*3/√2)^2 = 0 [so here next is where we use difference of squares (x^2 - y^2) = (x+y)*(x-y)] ( √2*x+(3/√2) + i*3/√2 ) * ( √2*x+(3/√2) - i*3/√2 ) = 0 [group the constant terms:] (√2*x + (3+i*3)/√2)*(√2*x + (3-i*3)/√2)=0 [you could be done here if you know that if (mx+n)=0 then x must equal -n/m, but if not, pull out a √2 from each factor:] √2*(x+(3+i*3)/2) * √2*(x+(3-i*3)/2)=0 [combine them:] 2*(x+(3+i*3)/2)*(x+(3-i*3)/2)=0 Then we just read off the solutions as normal: x={ -3/2 -(3/2)*i, -3/2+(3/2)*i } which are the same as in the video

[@Raizensen9415]

Best book for all basic concept for algebra

[@anandk9220]

Third method is the best of all time !!! 😘😘😘😘😘😘😘😘😘😘

[@Songfugel]

2nd one felt most comfortable, since it was close to the normal way to solve it

[@danieleferrara561]

Last one is the most interesting, thank you

[@nychan2939]

I love the completing square method since I was a child.

[@jgrigg1720]

I honestly love the third way.

[@user-my7ki4it3s]

First one. But the other two ways were a completely new for me, thanks 😅

[@sakib5766]

3rd method: when u r too crazy for a perfect square

[@krishanu-d1k]

First and second is best!!

[@sueyibaslanli3519]

Exactly, the third one

[@mathstutors9179]

7:35 What is the difference between “+or-“ and “-or+”

[@Crash-yp7ll]

If you have two or more of 'these' in eq, always use the top signs together or the bottom two signs together, but never the top and bottom of each together ...

[@stephenbeck7222]

No difference if you only have one of them in the equation. Sometimes we write, for example, the formulas for sum and difference for cosines with a +- on one side and a -+ on the other side to indicate when one side is adding then the other side is subtracting and vice versa.

[@Crash-yp7ll]

Agree ... - '/ ' always confusing - Doesn't totally mean '+ or -' but maybe better as '+ and -, one at a time, and in that order' ...

[@johnbiluke8406]

I like the Gatorade bottles in the background.

[@reda4495]

Blackpenredpen >>>> any other math's teacher

[@durgeshverma8131]

Hey bro superior literally quite shocked to see the way you solved I'm in 11th and just now studies complex number with quadratic equations.😃😄😄 LOVE from India

[@simdriver6797]

Mind blowing in t-3, t-2, t-1... now!

[@aamirakhtar5913]

We love your explanation 💘😻💜💛💚🧡 . Can you please tell us how you explain. When ever I try to explain any maths problem literally no one got it. You are awesome 👌

[@in-ty8vb]

I bet your explanation is good,they just don't know math

[@electrocode4095]

There is one another method (How you can forget Shri Dharacharya's Formula 🤨) ax² + bx + c = 0 you can write x =( - b ⨦ √ (b² - 4*a*c) )/(2*a)

[@flash.the.dash765]

Also known as the quadratic formula

[@flash.the.dash765]

Apparently there are 3 ways

[@AliKhanMaths]

Well-remembered! But that's just a derivation of the conventional completing the square method, so I don't think it would count as another method of completing the square.

[@electrocode4095]

@Python Project Academy It can be able to make square just do some math for it i.e. after rearranging of variables and constant it would be (x+ b/2a)² = (b² - 4*a*c )/4*a²

[@jimbobago]

This formula is simply the generic result from using completing the square on ax^2 + bx + c = 0.

[@skyjumper4097]

1:00 where do the 3x go?

[@nitish2799]

1:31 How can a square of any no be -ve?

[@siralanturing9103]

the cheeky smile when black pen says +/- but red pen says -/+

[@alpcanakaydn6986]

additional soluiton: if we multiply each side by 2 it becomes 4x²+12x+18=0 if we complete the perfect square it becomes (2x+3)²+9=0

[@Leeanne750]

Multiplying by "4a" is safe. (In this case, 8). It will work all the time, not only in this example. ax^2 + bx +c =0 4a^2x^2 + 4abx = -4ac (2ax + b)^2 = -4ac - b^2 .........

[@alpcanakaydn6986]

@@Leeanne750 It works just fine,but my goal is to find other solutions in the question

[@angelofelisimolacruz1462]

@@Leeanne750 Thanks for this approach.

[@muhammadnaufalzaky1326]

Please, upload a video about the del operator, that thing in the vector calculus

[@carultch]

The del operator is a vector of partial differential operators. When applied to the next function, it is applied in a way that is analogous to either scalar multiplication, dot products, or cross products. Apply del to scalar field f(x,y,z), analogous to scalar multiplication, and we get the gradient of the scalar field. The gradient is a vector field of partial derivatives of the scalar field, which indicates the direction and magnitude of steepest ascent. del f(x,y,z) = Apply del dot to a vector field F(x,y,z), analogous to the dot product, and we get what we call the divergence. This is a measure of sources and sinks in a vector field. An application of this, is Gauss's law. del dot F = dF/dx + dF/dy + dF/dz Apply del cross to a vector field F(x,y,z), analogous to a cross product, and we get what we call curl. This is a measure of the rotationality of a vector field, and a vector field with a curl of zero is called a conservative vector field. A conservative vector field is the gradient of a scalar function. The curl of a vector field is itself a vector field, that will be perpendicular at every point. It indicates that line integrals around closed loops of a vector field are non-zero. An application of curl is Faraday's law of induction and Ampere/Maxwell's law of current and magnetic fields.

[@MonzurulHoque3141]

& That's why I love Mathematics😍😍🤷😁😊😊

[@migssir]

Your 3rd solution is cool.

[@СвободныйУголок]

5:10 incredible methods, but one thing I did not understand, why multiply by 8 in the 2nd method? can be multiplied by 2 and get the same result without huge calculations 2(2х^2+6х+9)=0 4х^2+12x+()^2=-18+()^2 12=2*2*3 (2x+3)^2=-18+9 2x+3=+-r(-9) x=(-3+-3i)/2

[@dog.3162]

so it works consistently probably

[@mathysicssaransh7939]

Can u make a video explaining the graphical meaning of integration by substitution...more specifically .. integration by trigonometric substitution..?

[@carultch]

It means that you can recognize the integrand, as part of a term made from the Pythagorean theorem, in order to re-write a complicated term as a trig function that you can integrate. For instance: integral of dx/sqrt(1 - x^2) Draw a right triangle with (1-x^2) as one of its legs The remaining side will be x and the hypotenuse will be 1 This enables you to define an angle on this triangle, in order for 1/sqrt(1-x^2) to be a trigonometric ratio of the angle you define.

[@bhgtree]

"don't like fractions......when they see fractions 90% of them freak out." And thats only the teachers :)

[@kokichioma1408]

I'm new subscriber here.

[@dr.rahulgupta7573]

Sir (( d/dx) LHS )^2 = D , where D is discriminant of L HS function . This gives the answer immediately. i.e. (4x+6)^2 = -- 36 or 4x+6 = +6i or 4x +6 = --6i .

[@mokouf3]

3rd way is cleanest.

[@EynkiYoom]

Damn. Nice!!!!

[@glennn.micarandayo8657]

Just curious of the formula table on the background

[@daiyousei.1586]

This is so efficient holy sht Method 2 is my fav

[@ericthecuberspeedcuber]

5:46 "Wouldn't it be nice"

[@denilsoncosta31415]

Suggest 4th method to replace x by y-b/2a which is also cool.

[@qureshisiddig9274]

How amazing is that you keep superise me wow

[@sanadsingh7104]

Sir, can you tell me how to find the length of curve of sin x from 0 to pi

[@3r3nite98]

This is very cool.

[@noreldenzenky1527]

perfect

[@zerilioner639]

Hello, I have one math question:Find the equation of the tangent to the curve y =cos^2⁡ x when x=π/3 . Your solution should use radian measure. I kind of figured out m=-√3/2and y=1/4. Is the equation how to solve? Is it 1/4=√3/2(x-π/3)+y? Can you explain this question?

[@FaerieDragonZook]

Since there is a constant term, x=0 is not a solution. Thus, you can divide both sides by x^2. Then make the substitution y = 1/x, and do complete the squares in y. This makes the third way look like the first

[@nory2441]

Hey bprp love from India nice videos can you plz make 100 limits in 1 take plz I commented before but u din't see

[@willbishop1355]

I do really like the 3rd way, although as you said it won't always work.

[@PeetaN]

I prefer the first that we can use everytime :)

[@GDPlainA]

7:21 I cant unsee that -+ looking like a villager

[@prashanthkumar0]

all the methods were good but i still like quadratic formula :P

[@helloitsme3958]

hey bro i need help 1/a+a=b find 1/a^n+a^n in term b thanks

[@ouidadelbojaddaini1432]

How can we thank you?

[@professoraxolotl8960]

3:58 I'm a math teacher and the same thing happens to me quite a lot, I want to ask for something but I say the answer hahaha

[@eliasmazhukin2009]

In fact, way #2 is the way people were doing it in India in the 11th century!

[@johndanielvillanueva8604]

Awesome

[@GDPlainA]

the 2nd way is basically going from the quadratic formula to the base form

[@StephanvanIngen]

Beautiful

[@BlessingIduh]

I prefer the third way

[@omograbi]

There was a forth way you mentioned in other vids where you reduced the equation in a table and then multiplied the contents of the table.

[@robertveith6383]

*@ blackpenredpen* -- Treat the constant 3 as the variable from the start: 9 + 6x + 2x^2 = 0 ---> (3)^2 + 2x(3) + ______ = -2x^2 + ______ ---> (3)^2 + 2x(3) + x^2 = -2x^2 + x^2 ---> (3 + x)^2 = -x^2. Then, proceed as you did with the rest of Method 3.

[@particleonazock2246]

Just if you didn't know: bprp drilled a square peg in the round hole of his Pokeball. This enabled him to complete the square, in his niche field of mathematics.

[@DownDance]

I like that you say with multiply everybody... Like these are humans

[@thengodpbot7892]

can you do IMO 2021

[@arcader30]

4th way, use QUADRATIC FORMULA!! 😁😁

[@jimbobago]

But the only reason the quadratic formula exists is ... completing the square ... which is the point of the video.

[@blandon93]

Is there a way where one of x is equal to i or -i? I calculated one imaginary root which is close to -i (like -0,98i), I wonder if we ever can achieve whole i number.

[@flash.the.dash765]

The letter 'i'is basically an imaginary number since its not possible to solve the negative root. However if you write it as '-i' it'll simply cancels the negatives of root and the letter i and results in positive root

[@MohammadIbrahim-sq1xn]

=(x+i)(x+k) =x²+x(i+k)+ ik

[@arshadalam]

Simply use quadratic formula

[@marbangens]

the 3rd way, because 1+1 is fun :)

[@matejcataric2259]

I love math4fun and I love bprp and I love math

[@aryangupta7092]

Can u find the remainder when 17^(63) is divided by 1000

[@aashsyed1277]

Cool video! Thanks so much for these joyful moments ❤️nice work!

[@egorsuzopov2419]

Good day. Here in Russia we solve that equations in school way with Discriminant, which is: D = b^2 - 4ac. And the x1 & x2 = (- b ± i*√D)/2a. Much faster!

[@chitramc7238]

Same

[@robertveith6383]

Write 2a inside of parentheses when it is in the denominator.

[@thengodpbot7892]

Do you know general way to complete cube equation

[@XAHMED5A]

prove that from L.H.S (secA.secB.cscA.cscB)/(cscA.cscB - secA.secB) = sec(A+B)

[@pribato]

Are you from Singapore?

[@aweebthatlovesmath4220]

China

[@rikthecuber]

DO SOME DIFFERENTIAL EQNS BPRP!

[@blackpenredpen]

What's a differential equation? jk, here's the diff eq marathon: kzbin.info/www/bejne/m17Ghaydg8d4i6c

[@durgeshverma8131]

He means to use calculus method.

[@klausbuda7922]

I would use the quadratic equation to solve it

[@jimbobago]

But you need completing the square to develop the quadratic formula in the first place.

[@GreenMeansGOF]

I disagree with method 3. Square root of x^2 is the absolute value when x is real. However, x is not necessarily real and is actually complex and non-real. I do not think sqrt(x^2)=+/-x is valid. I could be wrong, however. And yes, I acknowledge that you arrive at the correct answer but that does not mean that the method is valid.

[@MathNerd1729]

Well, the square roots of -x² could be ix or -ix. The principal root could be either one of these, yet because of the ±, both possibilities are covered! :)

[@simplevc2980]

First way

[@DudsO_o]

although hating fractions, I prefer the first one

[@chocolateangel8743]

If you hate fractions, you should check out the fraction series that Dr. James Tanton has on his channel. He's a mathematics educator and researcher that's big on teaching the mathematical logic and concepts behind things. He believes that the more a student understands, the less he or she has to memorize. He deals with fractions differently than most people.

[@holyshit922]

For cubic equation ------------------------------------------ In book written in my native language in XVIII century there is following way Let's start from equation x^3+px+q=0 Move terms with x and constant to the other side x^3=-px-q Let's complete cube with free variable x^3+3x^2z+3xz^2+z^3=3x^2z+3xz^2+z^3-px-q (x+z)^3=(3xz+3z^2-p)x+z^3-q Now suppose that 3xz+3z^2-p=0 3z(x+z)=p x+z=p/(3z) p^3/(27z^3)=z^3-q z^6-qz^3-p^3/27=0 and we have quadratic in z^3 Interesting way for cubic which I saw on math forum uses sum of cubes identity x^3+px+q=A(x+m)^3+B(x+n)^3 Expand these cubes , compare coefficients and you will get system of equations to solve

[@MathNerd1729]

I just found out how the last way generalizes! It's quite fun! :) Start with: ax² + bx + c = 0 (c ≠ 0) Move the ax² term: bx + c = -ax² Multiply by 4c: 4bcx + 4c² = -4acx² Add b²x² and factor: b²x² + 4bcx + 4c² = (b² - 4ac)x² (bx + 2c)² = (b² - 4ac)x² Take the square root: bx + 2c = ± √(b² - 4ac) x Solve for x and rationalize the denominator to get the usual Quadratic Formula! :) Footnote: Apologies to BPRP about my previous comment; I meant the last way and didn't know how to tell you I made a typo! Additional note: The formula we get before rationalizing the denominator is known as the citardauq formula (That's quadratic backwards!)

[@karma_kun9833]

(Odd+Odd+Odd=Even)? Can we prove that's right or not?

[@Amar-zw5md]

IIT Advance aspirants be like : We had few more ways left to solve that :)

[@crispyclips6268]

👍👍

[@Atreyaa499]

By simple quadratic formula 🤣😂😂🤣🤣

[@ruceblee969]

I heard that pop 3:24. Maybe go see a chiropractor...

[@arirooz6240]

Oh the 🦒 is concentrated at class. It's so cute. This is a importand method about trinomious. TK