How to complete the square (when solving quadratic equations)
ฝัง
- เผยแพร่เมื่อ 24 ก.ย. 2024
- Let's discuss completing the square method when we are solving quadratic equations! We learn how to solve quadratic equations in 9th grade algebra but when we cannot solve a quadratic equation by factoring, we will have to use either completing the square or the quadratic formula. In fact, there's a geometric meaning behind the term completing the "square".
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When should we use completing the square instead of the quadratic formula? 👇
th-cam.com/video/5Y3dVXigIBI/w-d-xo.html
What happens to the 25 that you added to the left hand side of the equation?
I have never seen this “box” approach to solving for X before. Pretty cool.
btw, the quadratic formula is derived using this. There's a video by "Mind Your Decisions". It's pretty old but explains it well.
Me neither, excellent video
i learned about completing the square in ordinary differential equations. the most confusing math class i've ever taken.
Get ready for PDEs@@jaspertyler4557
It's called an area model. You can use them to do quite a few things. When you understand (and get practice with them), they really allow you to conceptualize things, so there is less to memorize.
Awesome. Thanks for the history video as well. I can see based on this how geometry lead to algebra, and eventually conundrums like "+- √x" that lead to the use of the plot graph solutions & proofs, and eventually, calculus.
What a great video! Would love to see more of these historically contentious math terms!
Thank you so much!
this is the best math channel ever and made understanding the whole completing the square so easily. thank you so much for making these cool videos
This would have made it so much easier to conceptualize in school!
On point, not too much talking. Great video. Thank you
damn who knew that actually explaining the concept instead of just listing steps aimlessly would make me actually fucking learn this concept 😭😭😭 thank you
You are the best math TH-camr 😊
This is a neat way of doing it. Of course, you could always do it algebraically, by subtracting 24 from both sides, getting x^2+10x-24, which can be factored out to (x-2) and (x+12), giving us the answers of x=2 and x=-12.
Brilliant explanation. I'm so jealous of kids today -- and teachers today! -- who can get these great explanations and learning methods at home for free. This geometric demonstration reminded me of 3blue1brown's geometric treatments of linear algebra. So cool.
This was also visualised in my Mathematics B Edexcel International GCSE study text.
Hey, I love your videos! You actually helped me pass my maths exam with a random exercise, and I thank you alot!!
(keep the good work up, love ur channel🔥🔥💯)
Quadratic solution now kinda makes sense geometrically - it's just a question if I want to add or remove from x square
Thank you thank you sooooooooooooooo much you saved me from the exam
The equation i.e
((1/√(x!-1)+1/x^2)!
It surprisingly approaches to 0.999.
For x>2
lim
x→∞
I would really appreciate you if you check it and I would like to ask can this be constant which is mine?
I like to define perfect squares first and then you just use c=(b/2)^2 and see what's extra
i finally know why it is in fact called "complete the square"
Thank you, proffesor
can you please post the solution to sqrt(1/x^2 - 1/x^3) + sqrt(1/x - 1/x^3) = 1 without just squaring both side and making it very long.
Here is how I did it (it does use squaring both sides but it's not that long, don't worry :)) :
First, let t=1/x. Our equation becomes sqrt(t^2-t^3)+sqrt(t-t^3)=1. We will now make a series of algebraic manipulations:
Isolate sqrt(t^2-t^3): sqrt(t^2-t^3)=1-sqrt(t-t^3)
Square both sides: t^2-t^3=1+t-t^3-2sqrt(t-t^3)
Cancel the t^3 terms and isolate 2sqrt(t-t^3): t^2-t-1=-2sqrt(t-t^3)
Square both sides again: t^4-2t^3-t^2+2t+1=4t-4t^3
Move everything to the LHS: t^4+2t^3-t^2-2t+1=0
Notice our LHS looks a lot like t^4-2t^3-t^2+2t+1, which we know is equal to (t^2-t-1)^2 since we worked it out earlier. This motivates us to introduce the substitution t=-y. Our equation then becomes y^4-2y^3-y^2+2y+1=0, which factors as (y^2-y-1)^2=0, which is equivalent to y^2-y-1=0.
Solving this quadratic equation gives us y=(1+-sqrt(5))/2. Since t=-y=1/x (our substitutions from earlier), we have x=-1/y.
Therefore, x=-1/((1+-sqrt(5))/2)=-2/(1+-sqrt(5)).
Rationalizing the denominator gives x=-2/(1+-sqrt(5))*(1-+sqrt(5))/(1-+sqrt(5))=-2(1-+sqrt(5))/(-4)=(1+sqrt(5))/2.
Hence, x=(1+sqrt(5))/2 or x=(1-sqrt(5))/2.
However, we have to reject the second solution since it makes the second square root in the original equation a complex number.
Therefore, the only solution x=(1+sqrt(5))/2, which just so happens to be the golden ratio!
Interesting analysis
Very helpful, thanks
you are a genius!
Lovely. But I miss smth on the top line, the actual reasoning why our ancestors did this. Not only the left hand side of the original equation represent
an area, but the right also. The geometric object with area X^2 + the geometric object with area 10*X equals the geometric object with area 24.
Imagine you are an Egyptian geometer. To get algebraic solutions to area problems is his task.
And now I'm confused
im not confused any longer
why do we always assume that x is greater than the number?
Graphs are not gonna be 100% accurate. If x was < or > or = the number, then just draw the x accordingly.
This is just to get the idea of where completing the square comes from.
@@zemyaso When people draw (a±b)² sometimes they assume either a>b or a
It doesn’t matter. I could have done a smaller square first then a bigger one. 😃
@@bprpmathbasics Divide by 2 or multiply by ½?
... Und jetzt noch den Zusammenhang zwischen x=2 und der Abbildung... bzw x=-12 und der Abbildung 😮.....
! warning do not trust this guy !