- WELCOME TO SOLVING QUADRATIC EQUATIONS USING SQUARE ROOTS.
WE'LL BE SOLVING EQUATIONS IN THE FORM OF AX SQUARED + C = 0.
THIS IS A VERY SPECIFIC FORM WHERE THERE IS NO X TERM.
AND WE'LL BE SOLVING THESE EQUATIONS
USING THE SQUARE ROOT PROPERTY,
WHICH STATES THAT THE EQUATION U SQUARED = D,
WHERE D IS GREATER THAN ZERO HAS EXACTLY TWO SOLUTIONS
WHERE U IS EQUAL TO THE SQUARE ROOT OF D
AND U IS EQUAL TO THE NEGATIVE SQUARE ROOT OF D.
AND SOMETIMES THIS IS WRITTEN AS
U EQUALS PLUS OR MINUS THE SQUARE ROOT OF D.
SO WHAT THIS IS TRYING TO TELL US IS IF WE HAD U SQUARED = 36,
THEN U WOULD EQUAL POSITIVE SQUARE ROOT OF 36, WHICH IS 6
AND U WOULD ALSO EQUAL NEGATIVE SQUARE ROOT OF 36,
WHICH IS EQUAL TO -6.
REMEMBER IF WE SQUARE -6, WE STILL GET POSITIVE 36.
SO WE'LL HAVE A POSITIVE AND A NEGATIVE SOLUTION.
AND NOTICE THAT IF D IS NEGATIVE OR LESS THAN ZERO
THAT THERE ARE STILL TWO SOLUTIONS,
IT'S JUST NOW THAT WE HAVE IMAGINARY SOLUTIONS.
LET'S GO AHEAD AND TAKE A LOOK AT SOME EXAMPLES.
SO EVEN THOUGH NUMBER 1 IS FACTORABLE
AS A DIFFERENCE OF SQUARES,
WE'RE GOING TO GO AHEAD AND SOLVE THIS USING
THE SQUARE ROOT PROPERTY.
SO LET'S GO AHEAD AND ADD 9 TO BOTH SIDES.
SO WE'LL HAVE X SQUARED = 9.
NOW IF WE HAVE X SQUARED AND WE WANT X,
WE CAN TAKE THE SQUARE ROOT OF BOTH SIDES OF THE EQUATION,
AS LONG AS WE REMEMBER TO INCLUDE A PLUS OR MINUS
TO OBTAIN THE TWO SOLUTIONS.
SO WE HAVE X EQUALS PLUS OR MINUS THE SQUARE ROOT OF 9,
WHICH IS 3.
AND IF WE WANTED TO, WE CAN LIST THIS AS X = 3 AND X = -3,
BUT OFTEN YOU'LL SEE THE ANSWER WRITTEN IN THIS FORM
TO REPRESENT BOTH SOLUTIONS.
OKAY, ON NUMBER TWO, NOTICE THIS IS NOT FACTORABLE
BECAUSE WE HAVE A SUM OF SQUARES,
BUT WE CAN USE THE SQUARE ROOT PROPERTY TO SOLVE THIS,
SO WE'LL SUBTRACT 64 ON BOTH SIDES.
SO WE HAVE X SQUARED = -64.
NOW TO GET X FROM X SQUARED,
WE'LL TAKE THE SQUARE ROOT OF BOTH SIDES
AND AGAIN PLUS OR MINUS.
SO WE'LL HAVE X EQUALS
PLUS OR MINUS THE SQUARE ROOT OF -1 x 64.
WELL, THE SQUARE ROOT OF -1 WOULD BE "I",
THE SQUARE ROOT OF 64 WOULD BE 8.
SO WE HAVE X = +/- 8I OR X = 8I AND X = -8I.
LET'S GO AHEAD AND TAKE A LOOK AT A COUPLE MORE OF THESE.
ON NUMBER THREE, AGAIN WE'RE TRYING TO ISOLATE THE X SQUARED
SO WE'LL ADD 64 TO BOTH SIDES.
SO WE HAVE 2X SQUARED = 64.
NOW IN ORDER TO ISOLATE THE X SQUARED,
WE STILL HAVE TO DIVIDE BY 2 IN THIS PROBLEM.
SO WE'LL HAVE X SQUARED = 32
AND NOW WE CAN SQUARE ROOT BOTH SIDES OF THE EQUATION,
PLUS OR MINUS THERE.
SO WE HAVE X EQUALS PLUS OR MINUS THE SQUARE ROOT OF 32,
BUT 32 IS EQUAL TO 16 x 2, 16 IS A PERFECT SQUARE,
SO WE HAVE X = +/- 4 SQUARE ROOT OF 2.
AND LET'S TAKE A LOOK AT ONE MORE.
AGAIN WE NEED TO ISOLATE THE X SQUARED FIRST,
SO WE'LL GO AHEAD AND SUBTRACT 7 ON BOTH SIDES.
SO WE HAVE 3X SQUARED = -7.
NOW WE NEED TO DIVIDE BY 3.
SO WE HAVE X SQUARED = NEGATIVE SEVEN-THIRDS.
NOW WE'LL SQUARE ROOT BOTH SIDES OF THE EQUATION
AND THERE'S OUR TWO SOLUTIONS.
SO WE HAVE X EQUALS PLUS OR MINUS--
REMEMBER WE CAN'T LEAVE A FRACTION
UNDERNEATH THE SQUARE ROOT,
SO WE COULD REWRITE THIS AS THE SQUARE ROOT -7/SQUARE ROOT OF 3,
BUT NOW WE CAN'T HAVE A SQUARE ROOT IN OUR DENOMINATOR
SO WE'LL MULTIPLY THE DENOMINATOR AND THE NUMERATOR
BY THE SQUARE ROOT OF 3.
SO WE'LL HAVE X EQUALS PLUS OR MINUS SQUARE ROOT OF--
WELL, THIS WOULD BE -21 OR -1 x 21
ALL OVER THE SQUARE ROOT OF 3 x 3.
SO WE'RE GOING TO HAVE +/- "I" SQUARE ROOT OF 21 ALL OVER 3.
AGAIN, NOTICE WE HAVE A POSITIVE AND A NEGATIVE SOLUTION.
OKAY, I HOPE YOU FOUND THIS VIDEO HELPFUL.
THANK YOU FOR WATCHING.