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- 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.