Skip the Quadratic Formula and Solve Tricky Quadratic Equations with Completing the Square Instead

Learn how to use Completing the Square to Solve Quadratic Equations an avoid the cumbersome calculations of the Quadratic Formula. Learn a simple way to complete the square and then learn how to solve Quadratic Equations from there.

[@pcmjpcmj6854]

Now I understand, when you said 2 under the square root means 1 outside the square root you meant the two 7's. I was confused for a few moments. Love the videos. Thank You. Would this method work if the middle term was just x?

[@helpwithmathing]

@pcmjpcmj6854 Ah, thank you for letting me know that was a confusing way to phrase that. I'll be more specific in future videos! And yes, you can use this method if x has no coefficient, you will just have some fractions to work with.

[@gusdenver]

Completing the square is a great method for understanding the principle. But, real world data almost never comes in a form where completing the square is possible. The values aren't integers or simple ratios. Plan on using the quadratic formula most of the time.

[@kerryrus]

You can use the complete the square method for any quadratic, I believe, even complex solutions.

[@chrishelbling3879]

I saw a video a while back, where the dude literally chopped up pieces of construction paper, to create a square & complete a missing corner. An algebra/geometry mash-up, but it made intuitive sense.

[@kimba381]

You CAN, but you wouldn't. In fact, unless a is 1, and b is even, I don't.

[@helpwithmathing]

This discussion is making me incredibly happy. Thanks so much everyone for chiming in. @gusdenver, I agree: real world data wouldn't be an obvious candidate for Completing the Square. For messier data, I would probably choose to use my graphing calculator and solve visually through tracing, or through the "calc" function of the calculator. And of course, the quadratic formula will get you there every time.

[@bigdog3628]

give me an example please. My theory is the vertex form, which is what this essentially is, is the only thing needed provided it is a quadratic equation and can't be solved by factoring.

[@rainerinedinburgh5807]

That's all well and good, but I find completing the square more complicated than just using the quadratic formula. The formula is basically a shortcut to avoid having to go through the rigmarole of completing the square yourself. It has already done it for you!

[@helpwithmathing]

One of the things I love the most is how everyone has different ways of learning and different things they find difficult and easy! I'm so happy for you that you enjoy the elegance of a formula. I myself enjoy the elegance of completing the square. It makes it clearer to me WHY to take the steps I'm taking than the formula does, and many students find things easier to remember when they can understand why they are doing them. AND, you just inspired a video for my Brainteaser PlayList: deriving the Quadratic Formula. Thanks for the comment and the inspiration!

[@rainerinedinburgh5807]

@@helpwithmathing Yes, it's great that there are often several different ways of solving a problem. Only in this case the two ways (completing the square and using the standard formula) are essentially the same way, since the formula is simple to derive by completing the square in the general case (without using actual numbers for the coefficients a, b, and c). I agree that there is a lot to be said for remembering how to get to answers from scratch rather than memorizing formulas. On the other hand, sometimes formulas are good, especially when the problem they are designed for occurs very often.

[@helpwithmathing]

Totally agree with everything you just wrote! Now, shh, don't tell, but this video is primarily aimed at my middle school/ early high school students who will soon encounter conics and the need to complete the square to find the centers of ellipses and hyperbolas and to convert quadratics from standard form into vertex form. This video's purpose is to plant this technique in their brains now to make their future mathing easier. Shhh, don't tell.

[@chrishelbling3879]

@@helpwithmathing 8th & 9th-graders do conics? Wow.

[@bigdog3628]

it is just vertex form it is super easy to remember and a lot less "clunky" than that quadratic formula.

[@charlesokuom8747]

Well explained

[@helpwithmathing]

Thank you, and thanks for watching!

[@bigdog3628]

this is vertex form essentially. Vertex form for a quadratic formula is y = a(x-h)² +k where h = 1/2(-b/a) and k = -a(h²) + c. the a b and c terms are the same as they are in the standard quadratic form of y= ax² + bx + c. so apply all this to the equation x² - 20x + 2 =o. We rewrite this as y = x² - 20x + 2. (if we set all the x terms to o we see the y intercept is 2. This is a bonus step) now we use our formula above: h = 1/2(20/1) this of course is 20/2 which is 10 k = -1(10²) + 2. this gives me -100 + 2 which is -98. (remember if there is no coefficient before a term then assume it is 1) This gives us y = (x-10)² - 98 now solve for x with the following formulas x = h + √(-k/a) and x = h - √(-k/a) [the ( ) are used to show that the √ applies to both the -k and a term] This gives us x = 10 + √98 and x = 10 - √98 (the a term is a 1 so we can ignore it here) √98 is not a perfect square. 49 * 2 also equals 98 and 49 is a square of 7 so the square root of 98 is 7√2 This means that the solution set of x is [10+7√2, 10-7√2] this and the quadratic formula are the two ways I learned how to solve quadratic equations. You just need it in the form of ax² + bx + c = 0 then you have enough information to convert it to vertex form. A lot of work shown above, but if I wasn't explaining each and every single thing it would of taken maybe 10 seconds to do.

[@helpwithmathing]

Great!

[@chrishelbling3879]

Psst, check the denominator when you wrote the formula.

[@helpwithmathing]

Eek! That'll teach me to film a video before coffee! I'll figure out how to edit that. Thanks for the heads up!

[@helpwithmathing]

OK!! Edited the brain fart right out! Thanks for having my back, and for being kind about it!

[@bigdog3628]

@@helpwithmathing lol it happens to the best of us

[@helpwithmathing]

@@bigdog3628 Thanks!

[@MFisher7346]

Completing the square is rather complicated and I can never remember how to do it. I could use the formula in my sleep.

[@helpwithmathing]

@MFisher7346 you are correct that the formula will get you there every time. Thanks for watching and interacting!

[@williamBryan-k2e]

this demonstrates an issue that I have had - why does it have to be 0 on right hand side. Well - Now I know - if you are dealing with square -there are not mutiple roots. so you do not have to have 0 on right. but if there are multiple roots, then it has to be zero on right - so that the equation can be set correctly ( if not zero - that raises too many possible answers ). Ok, got it.

[@helpwithmathing]

I might phrase this slightly differently: if you do not have a middle term (and x term, as opposed to an x^2 or a constant) then you can get your constant to the right side and take the square root of both sides (you will still end up with a plus or minus answer on that right side). If you have a middle term, you need to get a zero on the right side of the equation to be able to apply the rule of zero once you factor (the only way two things can multiply to zero is if one or the other of them is zero)