- WE WANT TO SOLVE THE QUADRATIC
EQUATION 2X SQUARED - 4X - 3 = 0
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.
SO LOOKING AT OUR EQUATION
WE NEED TO RECOGNIZE
THAT "A" = 2, B = -4,
DUE TO THE SUBTRACTION,
AND C = -3.
SO NOW WE'LL PERFORM
SUBSTITUTION
INTO THE QUADRATIC FORMULA.
SO WE'LL HAVE X = -B,
WHICH IS GOING TO BE -4 + OR -
THE SQUARE ROOT OF B SQUARED,
WHICH IS -4 SQUARED - 4 x A x C,
WHERE "A" IS 2 AND C IS -3.
AND ALL OF THIS IS OVER 2 x A
OR 2 x 2.
NOTICE HOW IN THIS FIRST STEP WE
ONLY PERFORM THE SUBSTITUTION,
AND NOW WE'LL BEGIN TO SIMPLIFY.
SO WE HAVE X = 4 + OR -
THE SQUARE ROOT.
-4 SQUARED IS 16, AND HERE WE
CAN THINK OF THIS AS -4 x 2,
THAT'S -8 x -3, THAT'S 24.
SO WE HAVE 24, ALL OVER 2 x 2,
WHICH IS EQUAL TO 4.
SO WE HAVE X = 4 + OR -
SQUARE ROOT OF 40/4.
THE NEXT STEP IS TO SIMPLIFY
THE SQUARE ROOT OF 40.
WE'RE LOOKING FOR
PERFECT SQUARE FACTORS OF 40.
WELL, 40 = 4 x 10.
FOUR IS A PERFECT SQUARE.
THE SQUARE ROOT OF 4 = 2,
SO THIS SIMPLIFIES TO
2 SQUARE ROOT 10.
SO NOW WE HAVE X = 4 + OR -
2 SQUARE 10/4.
AND NOW WE NEED TO CONTINUE
SIMPLIFYING,
BUT WE NEED TO BE CAREFUL.
WE CANNOT SIMPLIFY
THESE TWO FOURS,
BECAUSE WE CANNOT SIMPLIFY
ACROSS ADDITION OR SUBTRACTION.
SO THERE ARE TWO WAYS
TO SIMPLIFY THIS CORRECTLY.
ONE WAY, SINCE WE'RE DIVIDING
BY A BINOMIAL,
WOULD BE TO DIVIDE THE TERMS
IN THE NUMERATOR BY 4.
WE CAN WRITE THIS AS 4/4 + OR -
2 SQUARE ROOT OF 10/4,
AND THEN WE SIMPLIFY
EACH FRACTION INDIVIDUALLY.
WE HAVE X = 1 + OR -, AND HERE
THE 2 AND THE 4 SIMPLIFY.
THIS SIMPLIFIES TO 1,
AND THIS SIMPLIFIES TO 2.
SO WE HAVE SQUARE ROOT 10/2.
NOTICE WE HAVE TWO REAL
IRRATIONAL SOLUTIONS.
ONE SOLUTION IS
1 + SQUARE ROOT OF 10/2.
THE OTHER SOLUTION IS
1 - SQUARE ROOT OF 10/2.
BUT I DO WANT TO MENTION
IF WE SIMPLIFY THIS
IN A SLIGHTLY DIFFERENT WAY,
WHILE THE ANSWERS
WILL BE EQUIVALENT,
THE FORM WILL BE
SLIGHTLY DIFFERENT.
IF WE GO BACK UP
TO THIS FORM HERE,
X = 4 + OR - 2 SQUARE ROOT OF 10
DIVIDED BY 4,
AND FACTOR THE GREATEST COMMON
FACTOR OUT OF THE NUMERATOR,
WHICH WOULD BE 2,
WE WOULD HAVE 2 x THE QUANTITY 2
+ SQUARE ROOT 10 ALL OVER 4,
WHICH YOU CAN WRITE AS 2 x 2.
IN THIS FORM WE CAN SIMPLIFY
THIS 2 OVER THIS 2,
SO WE COULD ALSO EXPRESS
THE SOLUTION IN SIMPLEST FORM
AS X = 2 + OR - SQUARE ROOT 10
ALL OVER 2.
SO AGAIN, THIS FORM HERE IN BLUE
AND THIS FORM HERE IN GREEN
ARE IN SIMPLEST FORM,
EVEN THOUGH IT'S EXPRESSED
SLIGHTLY DIFFERENT.
PERSONALLY I PREFER THIS FORM
HERE,
BECAUSE BREAKING THIS UP
INTO TWO FRACTIONS
USUALLY LEADS TO LESS ERRORS
WHEN SIMPLIFYING.
OKAY, I HOPE YOU FOUND THIS
HELPFUL.
