BEST and WORST WAY to solve this Quadratic Equation

[@shazhu2455]

A similar way is to use Difference of two squares formula: (2x+1+4)(2x+1-4)=0. So 2x+5=0 or 2x-3=0 and solve for two solutions.

[@chewiebacka4377]

First thing that came to my mid as well.

[@johngreen3543]

That is the "best way" to go. It always pays to recognize the difference of 2 squares.

[@thomaspaszynski8888]

Another good way to do this is substitution. So, you let u = 2x + 1, that gets you u^2 - 16 = 0, factor this equation, gets you (u-4)(u+4) = 0, plug back 2x + 1 into the factored equation gets you (2x+1-4)(2x+1+4) = 0, simply it gets you (2x-3)(2x+5) = 0, then solve 2x-3 = 0 gets you x = 3/2 or 1.5. For the second solution, 2x+5 = 0 gets you x = -5/2 or -2.5

[@danielweir5867]

Yes -- that's the way I did it, substitute u = 2x + 1.

[@bstevens9831]

I've forgotten my technique but this particular problem makes another approach possible: first, recognize the equation says "something" minus 16 equals zero. We know that "something" must be 16 ( because 16 minus 16 is zero ) , so the result of squaring what's in parenthesis must equal 16. Going further we know that only 4 or minus 4 squared equals 16, so we know 2x + 1 equals 4 or minus 4. We also know that 2x must be 3 ( ignoring the second answer for now ) because after adding 1 it must total 4(-4 ). So the answer comes down to 2 times what equals 3 and the answer is 1.5 ( or 3/2 ) . Not a good way to approach this but some answers are possible without technique. But technique is required for difficult problems and the video author makes many valid points which I don't dispute. Thank you for the video.

[@knotwilg3596]

There are 3 approaches. 1) work out the square and then apply the formula for quadratic equations. 2) recognize 16 as a square and then 2a) do as you do: 2x+1 = +/- 4 2b) apply the formula for differences of squares: (2x+1 + 4) (2x + 1 - 4) = 0 Methods 2a and 2b rely on recognizing 16 as a square, which most people do. 25, 49, 64, 81, 100 and probably 121, 144, 169 ... are still recognized. But by 196, 225, (256), 289 ... we're probably into the area where people don't. Method 1) is heavy but systematic. Method 2a is elegant but particular. It's not really about "best" or "worst". It's applying your method to your abilities. An experienced mathematician will have a big toolset but also a high capacity to recognize the situation. A starting mathematician has neither. We should not teach them what to use in which situation, rather show there are different situations and toolsets, but all can come to the same conclusion.

[@danieldennis9831]

⇒x=3/2,-5/2 I assume that the point is to minimize the number of steps to minimize the places where errors could be included into the process. I would assume that the best way is to take a square root of both sides, which I did (2x+1)²=16 2x+1=±4 2x+1=-4 x=-5/2 2x+1=4 x=3/2 ⇒x=3/2,-5/2 My worst way would be to square 2x+1, include the -16 and then do the quadratic formula. I didn't even consider completing the square or factoring because it would take too many steps.

[@u-no-who]

So, what are those two answers applied to? What are we trying to figure out other than those two answers?

[@MrZako2000]

Difference of square is another way to do it, (2x+1)^2-4^2

[@alext8828]

I took the "worst" way. This is clever.

[@thomaspaszynski8888]

Worst way I feel is to solve it graphically. You may not get accurate answers and it is probably the longest. Did you solve it graphically Alex? What is the worst way?

[@sandraclaus5514]

Not everyone is good at note taking during class. I have a learning disability and will not get anything from a class if I am trying to listen and write notes.

[@jim55price]

First off, it wasn't hard at all to produce 4x^2 + 4x -15 = 0, factor that into (2x - 3) (2x + 5) = 0, and then produce the two desired solutions. 90 seconds tops. Easier, though, was to recognize the equation as the difference of two squares, produce (2x + 1) - 4 = 0 and (2x + 1) + 4 = 0, simplify to 2x - 3 = 0 and 2x + 5 = 0, and produce the two desired solutions then. 15 seconds tops. Cheers.

[@Arcadia61]

Without the explanation and extra steps he wrote for clarification, his method would take less than a minute.

[@terrygoyan]

Totally. I factored it in seconds. I think though that there are lots of these kinds of problems that cannot be solve easily that way!

[@thomaspaszynski8888]

Might be hard to do and you have to be lucky here, but another method is guess and check.

[@robertveith6383]

sqrt(25) just equals 5, not plus or minus 5.

[@lanisilvious7098]

Then how would you ever solve a quadratic problem (MUST have two solutions) that can be solved by taking square root of both sides (x^2=64)?

[@Icehso140]

x = 1.5...times 2 =3...plus 1 =4...squared is 16...minus 16 =0. Did it in my head in less than 30 seconds. Where's the hard part? Basically solving for the squ root of 16.

[@PreservationEnthusiast]

Likewise I solved it in my head in a few seconds. x=1.5 or -2.5 The square is already completed in fact I can almost see the answers just by looking at it. It's basically just a couple of linear equations. I can't imagine anyone would want to multiply out and use the quadratic formula on this. It's so counter intuitive!

[@aryusure1943]

I thought I took the best way which is to solve what's in the parenthesis thinking that it must be 4. If you don't forget that there are 2 solutions for a quadratic equation and yes, I made the same mistake again. :( At least I got half of the solution which is x = 3/2.

[@bartekskorupa]

BEST and WORST ways to teach math are: BEST way: Do it right, i.e: sqrt(25) = 5... WORST way: Do it wrong: sqrt(25) = +5 or - 5. This "plus or minus" thing happens ONLY!!!! when you have sqrt(a^2). sqrt(a^2)=|a| which can be written as sqrt(a^2)=a or sqrt(a^2)=-a.

[@benquinneyiii7941]

Remedial

[@zimbabweangamer7594]

x=1.5

[@zaratron]

Always skip the first useless 5 minutes of these videos

[@CitizenPerkins]

Haha, I usually skip to the spot where I see 4 checkmarks under the word 'notes'. 😁

[@patriciadorris1149]

🙋👍

[@TKRM2007]

Stop,saying “Ok”

[@mikehenry9672]

I had a professor that would do that all class after like every 2-3 sentences lmao 🤣 shit cracked me up

[@tygokoops1521]

I like that you use Windows 7 xD

[@richardbellam5]

All this talking you do is the main distraction! Just shut up and get to it!

[@bwanalikatuli1364]

You talk too much. Be straight up to the point

[@henkhu100]

PLEASE stop telling us that a square root can be negative! The square root of 16 is 4. Follow a course in math before making these kind of videos.

[@lanisilvious7098]

But it's a quadratic equation, it has to have two answers. If you only use 4, you get exactly one answer, which means there's a solution missing. So you MUST use the fact that there can be multiple roots, not only the so-called "principal square root." If you insist that you can't use -4 (or any other negative number) to set up the equations/formulas get the value of x, you will never be able to solve this (or other) quadratic equations,. especially any that are taking the square root of both sides. Try this: solve X^2=64 Take square root of both sides. X=8 or X=-8 (remember, it's quadratic so I MUST give two solutions) OR rewrite as x^2-64=0 Difference of two squares (X+__)(X-__)=0 (X+8)(X-8) So x+8 is zero or x-8 is zero or both are zero. X=8 , x=-8 Your way if the square root of 64 can only be 8, you would be unable to provide both zeros of the problem. (Show your work, on my example, prove me wrong)

[@henkhu100]

@@lanisilvious7098 with the square root I mean in my comment the value with the √ symbol. That is the symbol for the so called principal square root That is a non negative value by definition. √16 is 4 In his video he replaces √16 by -4 and +4 To solve x^2 = 16 it is not correct to write x= √16 and then say x= +/-4 Correct is x=+/-√16. And that gives 4 and -4 In his video he replaces √(x+2)^2. By x+ 2 That is wrong if x+2 is negative Correct is √(x+2)^2 = |x+2| because with the √ symbol you ask for the principle square root.

[@lanisilvious7098]

@@henkhu100 is the principal square root a RULE or a CONVENTION not universally accepted?

[@lanisilvious7098]

@@henkhu100 EDIT Actually, a number squared cannot be negative (any number squared will always be greater than or equal to zero), but -x times -x will always equal x^2 So a SQUARE cannot be negative but the root can be. Principal square root is a convention not universally accepted and that does not work everywhere in mathematics. Square root is not a function. Unless the problem so specifies

[@henkhu100]

@@lanisilvious7098 What is 0^2 (or 0 x 0). You say that the answer is not 0 because you say that a number squared cannot be zero. So what value do you have in mind for zero squared? A root can be negative indeed but the principal square root can't be negative by the definition of the principal root. Important is that we deal with the real numbers, because for complex numbers other rules are valid. But even for complex numbers there is a well defined principal square root, indicated with the radical symbol ✓. And that definition is universally accepted. And because ✓ is the symbol for the principal square root ✓16 is equal to 4 and not equal tot the other root of 16 which is -4. You can find the theory about square roots in many good books and on many websites.

[@grahamy3400]

Took about 20 seconds