Solving Quadratics by Completing the Square

Now we know how to factor polynomials, but sometimes that just won't work. We need a different method that might seem a little trickier, but it works every time! It's called completing the square, and it might sound like geometry, but it has to do with manipulating an expression to generate a polynomial that is a perfect square. What am I doing explaining it to you here, just watch the darn thing!

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[stupidmonkey]

I'm refreshing my math skills 5 years out of highschool. I've long wanted to get into higher maths but had some gaps in my understanding highschool subjects. This lesson has exposed my biggest deficiency thus far and I'm happy I've found an area to improve. It's a step in the right direction. Thank you Dave for this content. It's very succinct and easy to digest compared to the typical classroom form.

[White_Lotus]

a 7 min. video that took me 1 hr to take note, analyse, and understand...things are getting trickier now...whew, anyway thanks to you Prof. Dave. I'm refreshing my math to prepare something in the future after stopping school for a long time, and here you are shortening our journey.

[NickHope]

COMPREHENSION #3 SOLUTION:
4x^2 + x - 3 = 0
∴ 4x^2 + x = 3
Which is the same as 4x^2 + 1*x = 3
Divide everything by 4 to get x^2 on it's own: x^2 + (1/4)*x = 3/4
Half of 1/4 quantity squared = 1/8 * 1/8 = 1/64, to be added to both sides.
x^2 + (1/4)*x + 1/64 = 3/4 + 1/64 = 48/64 + 1/64 = 49/64
Express the trinomial on the left as a binomial squared, using the 1/8 from 2 lines above as the 2nd term in the binomial:
(x + 1/8)^2 = 49/64
∴ x + 1/8 = √(49/64) = √49/√64 = ±7/±8
∴ x = ±7/±8 - 1/8
We can ignore ±8 and just use positive 8, because that just changes the sign (±) of the fraction and we are doing that anyway by applying ± to the numerator, 7.
∴ x = 7/8 - 1/8 = 6/8 = 3/4 (ANSWER 1)
OR x = -7/8 - 1/8 = -8/8 = -1 (ANSWER 2)

[NickHope]

COMPREHENSION #4 SOLUTION
3x^2 - 2x - 1 = 0
∴ 3x^2 - 2x = 1
Divide everything by 3 to get x^2 on it's own: x^2 - (2/3)*x = 1/3
Half of 2/3 quantity squared = 1/3 * 1/3 = 1/9, to be added to both sides.
x^2 - (2/3)*x + 1/9 = 1/3 + 1/9 = 3/9 + 1/9 = 4/9
Express the trinomial on the left as a binomial squared, using the 1/3 from 2 lines above as the 2nd term in the binomial:
(x - 1/3)^2 = 4/9
∴ x - 1/3 = √(4/9) = √4/√9 = ±2/±3
∴ x = ±2/±3 + 1/3
We can ignore ±3 and just use positive 3, because that just changes the sign (±) of the fraction and we are doing that anyway by applying ± to the numerator, 2.
∴ x = 2/3 + 1/3 = 1 (ANSWER 1)
OR x = -2/3 + 1/3 = -1/3 (ANSWER 2)

[Adam-cnib]

it took me approx 3 days to completely understand this concept (and I had some self-doubts along the way). Very trickey but rewatching and searching for alternative sources eventually gets the job done. Onto next vid!! :)

[DevDevi]

This is some of the best maths explaining done in YouTube.

[c.p.holmes]

Thank you for always giving explanations as to why certain steps are done! Most people I've been taught by didn't do that, but I find it exceptionally useful because that's one way in which my mind works: if I don't have an explanation as to why something is done a certain way, it can be really confusing, because then I don't see how to get the same answer myself. I really appreciate your work, thank you again!

[modernkipling]

Got stumped on the last two questions, came back a few days later and it became much clearer. Good stuff Dave!

[Syntaxxed]

This is the best way anyone has ever explained completing the square to me. Came here from your video on calculating the area between two curves with integrals. You might be saving my grade!

[dvrmurthy]

I am so sad your math hasnt got much views

[natnaelbelayneh]

you're the best teacher i've ever had

[kripashankarshukla]

I think I should quit school and ask for professor Dave's address and take tuitions from him.

[yuukililith]

6:24 Remember... beggars can't be choosers.

[adamakii]

poor fractions, getting called ugly just for doing their jobs :(

[xosama]

Comprehension #3 and #4 Have three solutions:
Comprehension #3 = 1, -1, 3/4
Comprehension #4= 1, -1/3, 1/3
Check them out if you don't believe me!

[lufunonemakhavhani]

thank you very much u are a star prof u rock

[Danny-dfpo]

I thought half of 1/4 is 1/2, then when I really divided I laughed at myself, after spending 30 fu**** minutes

[ignacioleikis]

Spoilers!

For Q1: is 4 +/- root11 the same as +/- root11 + 4?
For Q2: is -2 +/- root7 the same as +/- root7 -2?

[mark]

The bottom questions took me several hours how the answers came to be because I got really hung up on the fractions. Also, the bottom left question took me a while to realize that 49 can be squared and, on the bottom right question, it never occurred to me that half of 2/3 is 1/3 instead I used 2/6 then square it.

[MiltosPol-qnzh]

Are you going to explain the factorial(!)???