- THE QUADRATIC FORMULA.
OUR GOAL IS TO USE 
THE QUADRATIC FORMULA
TO SOLVE QUADRATIC EQUATIONS
AND ALSO TO VERIFY 
THOSE SOLUTIONS GRAPHICALLY
WHEN POSSIBLE.
THE QUADRATIC FORMULA.
THE SOLUTIONS 
OF ANY QUADRATIC EQUATION,
AX SQUARED + BX + C = 0, 
WHERE "A" DOES NOT EQUAL 0
ARE GIVEN BY X = -B + OR - 
THE SQUARE ROOT OF B SQUARED
MINUS 4AC ALL/2A.
THE FORMULA IS NOT MAGIC.
IT'S BEEN DERIVED BY COMPLETING 
THE SQUARE ON THE EQUATION
AX SQUARED + BX + C = 0.
THIS IS SHOWN 
IN A DIFFERENT VIDEO.
GRAPHICALLY, THE -ZEROES 
OF A QUADRATIC FUNCTION OCCUR
WHERE Y OR F OF X = 0.
IF THE ZEROES OF A QUADRATIC 
FUNCTION ARE REAL,
THEY WILL BE THE X INTERCEPTS 
OF THE PARABOLA.
RECALL THAT TO FIND 
X INTERCEPTS, YOU SET Y = 0.
SO THERE'S A CONNECTION 
BETWEEN THE X INTERCEPTS
AND THE SOLUTIONS TO QUADRATIC 
EQUATIONS, AND HERE IT IS.
IF YOU ARE GIVEN F OF X 
= AX SQUARED + BX + C
AND YOU SET F OF X = 0,
YOU WOULD HAVE BASICALLY
A QUADRATIC EQUATION SET 
EQUAL TO 0.
SO THE SOLUTIONS 
TO THIS QUADRATIC EQUATION
WILL ALSO BE THE X INTERCEPTS 
AS LONG AS THEY ARE REAL VALUES.
LET'S TAKE A LOOK AT AN EXAMPLE.
WE WANT TO SOLVE 
-X SQUARED + 3X + 10 = 0.
CHECK THE SOLUTION GRAPHICALLY.
THIS IS FACTORABLE,
BUT WE ARE GOING TO SOLVE THIS 
USING THE QUADRATIC FORMULA.
FIRST THING I'M GOING TO DO 
IS IDENTIFY THE VALUES
OF "A," B AND C.
"A" = THE COEFFICIENT 
OF X SQUARED,
B = THE COEFFICIENT OF X, 
AND C = THE CONSTANT 10.
WE'RE GOING TO DO THE 
SUBSTITUTION INTO THE FORMULA
AND THEN BEGIN TO SIMPLIFY.
B SQUARED WOULD BE 3 SQUARED.
I'M NOT DOING ANY MATH 
THIS TIME.
I'M JUST DOING THE SUBSTITUTION.
NOW, LET'S BEGIN TO SIMPLIFY.
LET'S COME BACK 
TO THE SQUARE ROOT.
OKAY. THE NUMBER UNDERNEATH 
THE SQUARE ROOT IS CALLED,
"THE DISCRIMINATE."
LET'S SEE IF WE CAN FIGURE OUT 
WHAT'S THAT'S GOING TO BE.
3 SQUARED, OF COURSE, 
WOULD BE 9.
WE HAVE 9 - -40,
SO THAT WOULD BECOME 9 + 40 
OR 49.
AND IT HAPPENS THAT 49 
IS A PERFECT SQUARE,
SO THIS BECOMES -3 + OR - 7 
DIVIDED BY -2.
SO WE HAVE TWO SOLUTIONS.
THE FIRST SOLUTION,
X = 03 + 7 DIVIDED BY -2,
THAT WOULD GIVE US A VALUE 
OF -2.
AND OUR SECOND SOLUTION, X = -10 
DIVIDED BY -2 = 5.
SO OUR SOLUTIONS 
TO THIS QUADRATIC EQUATION
ARE X = -2 AND X = 5.
NOW GRAPHICALLY, IF WE LET 
F OF X EQUAL THIS QUADRATIC,
-2 AND +5 SHOULD BE 
OUR X INTERCEPTS.
LET'S VERIFY THAT.
TO SAVE SOME TIME, I'VE ALREADY 
TYPED IN THE QUADRATIC EQUATION
INTO Y1.
TAKE A LOOK AT THE GRAPH, 
AND THIS DOES VERIFY OUR WORK.
WE HAVE AN X INTERCEPT HERE 
OF -2 AND X INTERCEPT OF +5.
I COULD ALSO CHECK THE TABLE 
TO MAKE SURE THAT WHEN X IS 5,
I HAVE A Y VALUE OF 0.
AND WHEN X IS -2, 
I HAVE A Y VALUE OF 0.
OUR WORK IS DONE CORRECTLY.
LET'S TRY ONE MORE.
SOLVE 2X SQUARED - 4X - 3 = 0.
CHECK YOUR ANSWER GRAPHICALLY 
ON THE GRAPHING CALCULATOR.
OKAY, WE'RE GOING TO GO AHEAD 
AND IDENTIFY OUR VALUES
FOR "A," B AND C.
"A" = 2, B = -4, C = -3.
AGAIN, THE FIRST STEP, I'M JUST 
GOING TO DO THE SUBSTITUTION,
ALL/2 x "A," 
AND AGAIN, START SIMPLIFYING.
OKAY. SO -4 BECOMES +4 + LET'S 
COME BACK TO THE DISCRIMINATE.
OUR DENOMINATOR IS ALSO 4.
OKAY, ON THIS ONE, WE'RE GOING 
TO HAVE -4 SQUARED WHICH IS +16
MINUS -24 WHICH BECOMES +24,
SO 16 + 24 WOULD GIVE US 40.
WE NEED TO SIMPLIFY 
THE SQUARE ROOT.
NOW REMEMBER, 40, 
I COULD WRITE 40 AS 4 x 10
AND THE SQUARE ROOT OF 4
WOULD BE 2.
SO THIS WOULD BECOME 4 + 2 
SQUARE ROOT 10 DIVIDED BY 4.
NOW, IT'S VERY TEMPTING 
TO SIMPLIFY THESE TWO 4s,
BUT THAT WOULD BE INCORRECT.
I ALWAYS LIKE TO TAKE 
THE EXTRA STEP
AND WRITE THIS AS TWO SEPARATE 
FRACTIONS TO SIMPLIFY IT.
4/4 + OR - 2 SQUARE ROOT 
OF 10/4.
I'D SIMPLIFY THEM INDIVIDUALLY.
THIS WILL ELIMINATE ANY --  
SIMPLIFYING MISTAKES.
4/4 WILL GIVE US 1 + OR - 2/4, 
OF COURSE, SIMPLIFIED TO 1/2,
SO IT'S SQUARE ROOT OF 10 
DIVIDED BY 2.
SO OUR SOLUTIONS,
X = 1 + THE SQUARE ROOT OF 10 
DIVIDED BY 2
AND X = 1 - SQUARE ROOT OF 10 
DIVIDED BY 2.
NOW, IT WOULD BE VERY DIFFICULT 
TO TRY TO VERIFY THESE VALUES
AS X INTERCEPTS,
BECAUSE THEY'RE IRRATIONAL.
SO I'VE ALREADY TAKEN THE TIME 
TO CONVERT THEM INTO A DECIMAL,
AND THIS WOULD BE APPROXIMATELY 
2.58
AND THIS WOULD BE APPROXIMATELY 
-.58.
THESE ARE THE VALUES
WE WILL LOOK TO SEE
IF THESE ARE OUR X INTERCEPTS.
LET'S GO BACK 
TO THE GRAPHING CALCULATOR.
OKAY, I ALREADY HAVE 
THE FUNCTION TYPED INTO Y2,
SO WHAT I'M GOING TO DO 
IS GO OVER TO THE EQUAL SIGN
ON THE FIRST FUNCTION 
THAT WE JUST USED.
HIT "ENTER" TO TURN IT OFF,
AND HIT "ENTER" ON Y2 
TO TURN IT BACK ON,
AND I'M GOING TO GRAPH IT.
THIS LOOKS LIKE THIS 
IS SOMEWHERE BETWEEN -1 AND 0,
MAYBE -.58.
AND THIS ONE IS BETWEEN 2 AND 3, 
AGAIN, A GOOD SIGN.
LET'S JUST GO AHEAD AND VERIFY 
ONE OF THEM MORE SPECIFICALLY.
IF I HIT 2ND TRACE, 
BRINGS UP THE CALCULATION MENU,
AND I CAN FIND THE ZERO AND 
I CAN FIND ONE OF THE ZEROES.
IT'S GOING TO ASK ME TO GO THE 
LEFT SIDE OF ONE OF THE ZEROES.
LET'S VERIFY THE ONE
ON THE RIGHT.
I'M ON THE LEFT SIDE.
I'M GOING TO HIT ENTER AND MOVE 
TO THE RIGHT SIDE OF THAT 0,
HIT ENTER AND THEN ENTER AGAIN,
AND IT LOOKS LIKE 
THAT IT IS CORRECT.
THE 0 IS 2.58 OR APPROXIMATELY
WHICH IS ONE OF THE VALUES 
THAT WE FOUND.
I'LL LEAVE THE OTHER 
VERIFICATION TO YOU.
I HOPE THAT HELPS EXPLAIN 
HOW TO USE THE QUADRATIC FORMULA
AND HOW TO VERIFY IT GRAPHICALLY 
WHEN POSSIBLE.
