How To Prove The Quadratic Formula By Completing The Square - Free Educational videos for Students in K-12

How To Prove The Quadratic Formula By Completing The Square - By The Organic Chemistry Tutor

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00:00	in this video , we're going to talk about how
00:02	to derive the quadratic formula . Starting from this equation
00:08	, A X squared plus bx plus C Is equal
00:11	to zero . The quadratic equation . So how can
00:15	we derive this particular formula X is equal to negative
00:19	B plus or minus the square root of B squared
00:24	minus four A . C . All divided by two
00:27	way in order to derive that formula . What we
00:33	need to do is we need to solve for X
00:36	and we have to complete the square in order to
00:39	do so . So let's get rid of this to
00:44	make more space . The first thing I'm gonna do
00:49	is I'm going to take the constant term C and
00:51	move it to the other side of the equation .
00:54	So we're gonna have a X squared plus bx .
00:59	Leave A . Space equals negative C . Now the
01:05	next thing that we need to do is divide everything
01:08	by A . So they will cancel here , we're
01:14	going to have X squared plus B over a ,
01:19	times X is equal to negative C over A .
01:25	So now at this point we need to complete the
01:27	square in order to do that , we need to
01:31	take half of that coefficient B over A 1/2 times
01:36	b over a . Is B over two a .
01:41	So we're going to add that to both sides of
01:43	the equation , but we need to square this result
01:45	, so we're gonna add plus B over two A
01:50	squared . Whatever you do to the left side ,
01:53	you must also due to the right side , so
01:56	that the value on both sides of the equation remain
02:00	the same . So now at this point we need
02:10	to factor the perfect square . Try no meal that
02:13	we have on the left side . A quick and
02:16	simple way to factor a perfect square . Try no
02:18	meal is to follow these steps , it's going to
02:21	be X , whatever variable you see there and then
02:24	whatever sign you see here plus whatever is here before
02:29	its squared . So X plus B over two A
02:33	squared . That's a very quick and simple way to
02:37	factor a perfect square . Try no meal on the
02:39	right side , we can go ahead and square that
02:42	term . So it's gonna be B squared over to
02:46	a squared is going to be for a squared .
02:54	Now the next thing that we want to do is
02:55	we want to convert This expression from two fractions into
03:01	one fraction . And so to combine two fractions into
03:04	one , you need to get common denominators . Therefore
03:07	we're going to multiply The top and bottom of sea
03:10	over a by four a . So we're gonna have
03:16	X plus B over A . I mean , B
03:20	over two A squared , and that's going to be
03:26	equal to B squared over four A squared minus .
03:30	So we have four a . c over four A
03:33	squared . So now that we have common denominators ,
03:37	we can combine it into a single fraction . So
03:40	this is going to be b squared minus four A
03:43	. C . All over for a squared . Now
03:52	, what do you think is the next thing that
03:54	we need to do at this point ? Remember our
03:59	goal is to isolate X . We need to solve
04:02	for X . So what we need to do is
04:04	take the square root of both sides . The square
04:08	root and the square will cancel . And so we
04:11	no longer need to write the parentheses . So it's
04:13	simply going to be X plus B over to A
04:19	. And then that's going to be the square root
04:22	of B squared minus four A . C . Now
04:25	the square root of for a squared , we could
04:28	simplify that . The square root of four is to
04:31	the square root of a square is eight . Now
04:34	, because we took the square root of the right
04:35	side , we can add plus or minus . The
04:40	next thing we need to do is subtract wolf sides
04:43	by this term or simply move it from the left
04:46	side to the right side . It's positive on the
04:48	left side , but it's going to be negative on
04:50	the right side . So we're gonna have X is
04:58	equal to negative B over to a plus or minus
05:02	the square root of B squared minus four A C
05:06	over two A . Now that we have common denominators
05:12	, we can combine that into a single fraction .
05:17	So this is going to be negative B plus or
05:19	minus the square root of B squared minus four A
05:23	. C . All over two . A . And
05:27	so that's how you can prove or derive the quadratic
05:32	formula starting from the quadratic equation . The key is
05:37	to complete the square and then solve for X .

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