Impress your Math Teacher! Solve by Completing The Square

How to solve a quadratic equation by completing the square - step by step example.

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TC Math Notes

[davidschwartz]

Hi John. Didn't get my answer 100% correct! - I got my solution all the way to the very last stage, but omitted to put it all over 4. I recon that in a real test, I would have gained most of the marks, so we're getting there. Thanks for doing these videos and I wish when I was at Senior School (sort of the equivalent to High School in the UK) I'd had a teacher like yourself. Much respect to you John.

[arkanrb]

THANK YOU in all languages of the world
greetings from Yemen

[Tkkj]

Thank you for your in depth analysis of how to solve quadratic equation using completing the square methods.

[danieldennis]

My answer I got from completing the square was ⇒x=(-5±√33)/4
I admit that completing the square is my weakest skill in dealing with quadratic equation solving.
My first step was dividing the equation by 2 to make the A coefficient 1 (ax²+bx+c=0). Then move the constant to the other side.
x²+5/2x=1/2 Let's make it 10/4 as the B coefficient and make it 16ths on the other side and then add 25/16 to both sides (perfect square).
x²+10/4x+25/16=8/16+25/16 factor the left and simplify the right
(x+5/4)²=33/16 find square root of each side (remembering it is ± on the right)
x+5/4=±√33/4. move the 5/4 to the other side and simplify
x=(-5±√33)/4
Let's try the quadratic formula to check this.
x=(-b±√b²-4ac)/2
x=-5±√(25-4(2)(-1)/2(2)
x=(-5±√33)/4
This checks. So that is the answer.
But this took me forever -- I would have preferred the quadratic formula as this would have been much faster.

[Lisa-tnl]

You turned the left side of the equation into a perfect trinomial square. That's how it's done.👍

[larrywiener]

If I were to have to solve this equation I probably use the formula, but this is a good exercise in completing the square hand and understanding algebraic operations

[tomtke]

ever since learning the quadratic equation I forgot other methods!

so I searched internet and relearned completing the square.

2X^2 + 5X - 1 = 0
2X^2 + 5X = 1
X^2 + (5/2)X = 1/2
X^2 + (5/2)X + (25/16) = (1/2) + (25/16)
(X + (5/4))^2 = ((8/16) + (25/16))
(X + (5/4)) = sqrt(33/16)
(X + 5/4) = +/- (1/4)(sqrt(33))
X = -(5/4) +/-(1/4)(sqrt(33))
= (1/4)(-5 +/-(sqrt(33))
= -(5/4) +/- sqrt (33/16)

VERIFY
I'll use the quadratic equation...
i.e. ( -b +/- sqrt (b^2 -4ac))/2a
a = 2
b = 5
c = -1
so....
(-5 +/- sqrt (5^2 -(4)(2)(-1))/ (2)(2)
(-5 +/- sqrt (25 + 8))/4
(-5 +/- sqrt (33))/4
(-5/4 +/- sqrt (33/16) ✔️✔️✔️

or -5/4 = -1.25
sqrt(33/16) = sqrt(2.062) = 1.436

-1.25 +/- 1.436
-1.25 + 1.436 = 0.186 root1
-1.25 - 1.436 = -2.686 root2

VERIFY
2X^2 + 5X - 1 = 0
root.1 X = 0.186
(2)(0.186)^2 + (5)(0.186) - 1 =? 0
0.069 + 0.93 - 1 ?= 0
0.999 - 1 ?= 0 ✔️✔️✔️

root.2 X = -2.686
2X^2 + 5X - 1 = 0
(2)(-2.686)^2 + (5)(-2.686) - 1 ?= 0
14.429 - 13.43 - 1 ?= 0
0.999 - 1 ?= 0 ✔️✔️✔️

[nintendogamer]

Really helpful video! But the long intro was dragging on and on unnecesarily ;P

[elliotlambert]

When you remove the square root of 33 over 4 why does the 4 not also become + and - too.

[afre]

What you did here, is it actually to create the proof of the quadratic formula??

[JohnnyGuitar]

What an incredibly good teacher!!
Thank you so much John.

[dahcargo]

This is the first time that the answer makes no sense regarding how it was arrived at by the teacher. The rule is that whatever you do to one side of the equation, you must do to the other. I have know idea how the 5/2X is converted to 5/4 by in essence, 3 removing the X and multiplying by 2? That is a violation of basic math, the variable must be accounted for, yet in his explanation it is nonexistent. I got the first part easily: X squared + 5/2X = 1/2, but it goes into nonsense for me. I might need to "brush" up on basic algebra, but I'm a College graduate with a Degree in Biochemistry and Biology w/ emphasis on Integral Calculus. Please help, where have I gone wrong? Granted, I'm 67 and graduated in 1980. but still, this hurts!!

[WombatMan]

Technically the x² part doesn't need to be 1, any square number will do. I found the numbers much easier to deal with by doubling the original equation (2x² + 5x - 1 = 0) to get (4x² + 10x - 2 = 0) and proceeding the rest the same way. Meant I was dealing with numbers over 4 instead of numbers over 16.

[Eduardo-tqsk]

When its has lot of, exponents, roots in denominator, power : people has to know the properties 😅.

[francisdelpuech]

The quadratic formula would have been faster but the CT was required

[sawdustwoodchips]

Thanks for this. But, No explanation as to why you half the x coefficient. And square. Just like many demonstrations it glosses over the fundamentals of maths. Better to show this graphically and then show that the quadratic equation is derived from completing the square. Also no mention of the fact that you can use completing the square to find the turning point of the quadratic.

[johncharlton-yq]

I am going to go and drink some wobbly pops catch you tomorrow

[faukerconsulting]

☝🏻I would have enjoyed math more had the question AND answer then breakdown how you came to the conclusion...my brain seems to work backwards much like dyslexia but in its own recognition 🤷🏻‍♀️
💁🏻‍♀️My "struggle" would be (did i forget to mention my brain works in its own way and inaccurately🙄) :
2X × 1 = 2
2X × 2 = 4
2 + 4 = 6
6 - 5 = 1

[thomashawkinson]

Too long and complicated. Better to skip this and focus on other math tools.

[harrymatabal]

Waste of time. Use quadratic formula. Easy