- WE WANT TO SOLVE 
THE GIVEN QUADRATIC EQUATION
USING THE QUADRATIC FORMULA 
PROVIDED HERE IN RED,
WHERE "A" IS THE COEFFICIENT 
OF THE X SQUARED TERM,
B IS THE COEFFICIENT 
OF THE X TERM,
AND C IS THE CONSTANT TERM.
SO WE FIRST NEED TO RECOGNIZE 
THAT "A" = 2, B = 4, AND C = 5.
SO USING THESE VALUES 
WE'LL PERFORM SUBSTITUTION
INTO THE QUADRATIC FORMULA.
SO WE'LL HAVE X = -B, 
WHICH IS -4,
+ OR - THE SQUARE ROOT OF 
B SQUARED, WHICH IS 4 SQUARED,
- 4 x A x C, 
WHERE "A" IS 2 AND C IS 5.
WE'RE GOING TO DIVIDE ALL 
OF THIS BY 2 x A OR 2 x 2.
NOTICE HOW IN THE FIRST STEP WE 
DIDN'T PERFORM ANY CALCULATIONS.
WE JUST PERFORMED 
THE SUBSTITUTION.
AND NOW WE'LL BEGIN TO SIMPLIFY.
SO WE'LL HAVE -4 + OR - THE 
SQUARE ROOT OF 4 SQUARED IS 16.
AND WE'LL HAVE - 4 x 2 = 8 x 5 
= 40, SO - 40.
THE DENOMINATOR IS 4,
SO WE HAVE X = -4 + OR - THE 
SQUARE ROOT OF 16 - 40 = -24.
THIS IS ALL OVER 4.
AND NOW WE WANT TO SIMPLIFY 
THE SQUARE ROOT OF -24,
WHICH WILL SIMPLIFY 
TO AN IMAGINARY NUMBER.
SO THIS EQUATION IS GOING 
TO HAVE TWO COMPLEX SOLUTIONS.
SO THE SQUARE ROOT OF -24 = 
THE SQUARE ROOT OF -1 x 24.
AND NOW WE WANT TO FIND 
PERFECT SQUARE FACTORS OF 24.
THE ONLY PERFECT SQUARE FACTOR 
IS GOING TO BE 4,
SO WE CAN WRITE THIS AS 
-1 x 4 x 6.
SO THIS SIMPLIFIES TO THE 
SQUARE ROOT OF 4, WHICH IS 2,
THE SQUARE ROOT OF -1, 
WHICH IS "I",
AND THEN WE HAVE 
THE SQUARE ROOT OF 6.
SO WE HAVE X = -4 + OR - 2I 
SQUARE ROOT 6 DIVIDED BY 4.
AND WE NEED TO CONTINUE 
SIMPLIFYING HERE,
BUT WE DO NEED TO BE CAREFUL 
HERE,
BECAUSE WE CANNOT JUST SIMPLIFY 
THIS -4 WITH THIS 4.
WE CANNOT SIMPLIFY 
ACROSS ADDITION.
SO THERE'S A COUPLE OF WAYS 
OF SIMPLIFYING THIS.
ONE WAY, BECAUSE WE'RE DIVIDING 
BY A BINOMIAL,
IS TO WRITE THIS AS X = 
-4 DIVIDED BY 4
+ OR - 2I SQUARE ROOT OF 6 
DIVIDED BY 4.
SO WE DIVIDED EACH TERM 
IN THE NUMERATOR BY 4,
AND NOW WE SIMPLIFY AGAIN.
THIS WOULD BE -1 + OR -, 
THE 2 AND THE 4 SIMPLIFY,
SO WE HAVE "I" SQUARE ROOT 6 
ALL OVER 2.
SO AGAIN WE HAVE 
TWO COMPLEX SOLUTIONS.
ONE IS X = -1 + I SQUARE ROOT 6 
DIVIDED BY 2,
OR WE HAVE X = -1 - I 
SQUARE ROOT 6 DIVIDED BY 2.
BUT I DO WANT TO SHOW THAT 
IF WE SIMPLIFIED THIS
IN A SLIGHTLY DIFFERENT WAY,
WHILE THE ANSWERS 
WOULD BE THE SAME,
THEY'LL BE IN A SLIGHTLY 
DIFFERENT FORM.
IF WE START BACK WITH THIS FORM 
HERE,
X = -4 + OR - 2I SQUARE ROOT 6 
DIVIDED BY 4,
AND FACTOR OUT 
THE GREATEST COMMON FACTOR
FROM THE NUMERATOR,
WE WOULD HAVE 2 x THE QUANTITY 
-2 + OR - "I" SQUARE ROOT 6
DIVIDED BY 4, 
WHICH WE CAN WRITE AS 2 x 2.
AND NOW WE HAVE A COMMON FACTOR 
OF 2 HERE AND HERE
THAT WOULD SIMPLIFY OUT,
LEAVING US WITH X = -2 + OR - 
I SQUARE ROOT 6 ALL OVER 2.
SO IT'S IMPORTANT TO RECOGNIZE 
THAT THIS FORM HERE
AND THIS FORM HERE 
ARE EQUIVALENT,
AND THEREFORE 
WOULD BE ACCEPTABLE.
I PREFER THIS FORM HERE,
AND THEREFORE THIS IS THE REASON 
WHY I LISTED THE TWO SOLUTIONS
IN BLACK IN THIS FORM.
BUT WE COULD ALSO WRITE THIS AS 
TWO SOLUTIONS
IN A SIMILAR WAY THAT WE DID 
HERE IN BLACK.
OKAY, I HOPE YOU FOUND THIS 
HELPFUL.
