CHAPTER 22-6 QUADRATIC FORMULA

PROBLEM OF THE DAY

• IF AND THEN WHICH OF THE FOLLOWING IS THE SUM OF N AND M?

• A) -8

• B)-6

• C)-4

• D)6

• E)8

ANSWER TO THE PROBLEM OF THE DAY

• SOLUTION IS ANSWER A

OBJECTIVES

STUDENTS WILL BE ABLE TO :* SOLVE QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA.

* CLASSIFY ROOTS USING THE DISCRIMINANT.

QUADRATIC FORMULA

• YOU HAVE LEARNED SEVERAL METHODS FOR SOLVING QUADRATIC EQUATIONS: GRAPHING, MAKING TABLES, FACTORING, USING SQUARE ROOTS, AND COMPLETING THE SQUARE. ANOTHER METHOD IS TO USE THE QUADRATIC FORMULA, WHICH ALLOWS YOU TO SOLVE A QUADRATIC EQUATION IN STANDARD FORM.

HOW DOES THE QUADRATIC FORMULA WORKS

• BY COMPLETING THE SQUARE ON THE STANDARD FORM OF A QUADRATIC EQUATION, YOU CAN DETERMINE THE QUADRATIC FORMULA.

THE QUADRATIC FORMULA

Often, the simplest way to solve "ax2 + bx + c = 0" for the value of x is to factor the quadratic, set each factor equal to zero, and then solve each factor. But sometimes the quadratic is too messy, or it doesn't factor at all, or you just don't feel like factoring. While factoring may not always be successful, the Quadratic Formula can always find the solution.

EXAMPLE #1

• FIND THE ZEROS OF F(X)= 2X2 – 16X + 27 USING THE QUADRATIC FORMULA.

EXAMPLE #2

• FIND THE ZEROS OF F(X) = X2 + 3X – 7 USING THE QUADRATIC FORMULA.

EXAMPLE #3

• USE THE QUADRATIC FORMULA TO SOLVE X2 + 3X – 4 = 0

EXAMPLE #4

• FIND THE ZEROS OF F(X) = 4X2 + 3X + 2 USING THE QUADRATIC FORMULA.

EXAMPL#5

• FIND THE ZEROS OF G(X) = 3X2 – X + 8 USING THE QUADRATIC FORMULA.

STUDENT GUIDED PRACTICE

• DO PROBLEMS 1-8 FROM THE WORKSHEET USING QUADRATIC FORMULA

DISCRIMINANT

• WHAT IS THE DISCRIMINANT?

• THE DISCRIMINANT IS PART OF THE QUADRATIC FORMULA THAT YOU CAN USE TO DETERMINE THE NUMBER OF REAL ROOTS OF A QUADRATIC EQUATION.

TYPES OFDISCRIMINANT

EXAMPLE #6

• FIND THE TYPE AND NUMBER OF SOLUTIONS FOR THE EQUATION.

• X2 + 36 = 12X

• X2 – 12X + 36 = 0

• B2 – 4AC FIND THE DISCRIMINANT

• (–12)2 – 4(1)(36)

• 144 – 144 = 0

• SINCE B2 – 4AC = 0

• THE EQUATION HAS ONE DISTINCT REAL SOLUTION.

EXAMPLE #7

• FIND THE TYPE AND NUMBER OF SOLUTIONS FOR THE EQUATION.

• X2 + 40 = 12X

EXAMPLE#8

• FIND THE TYPE AND NUMBER OF SOLUTIONS FOR THE EQUATION.

• X2 – 4X = –4

STUDENT GUIDED PRACTICE

• FIND THE TYPE AND NUMBER OF SOLUTIONS FOR EACH EQUATION.

• A) X2 – 4X = –8

• B) X2 – 4X = 2

QUADRATIC FORMULA APPLICATIONS

• AN ATHLETE ON A TRACK TEAM THROWS A SHOT PUT. THE HEIGHT Y OF THE SHOT PUT IN FEET T SECONDS AFTER IT IS THROWN IS MODELED BY Y = –16T2 + 24.6T + 6.5. THE HORIZONTAL DISTANCE X IN BETWEEN THE ATHLETE AND THE SHOT PUT IS MODELED BY X = 29.3T. TO THE NEAREST FOOT, HOW FAR DOES THE SHOT PUT LAND FROM THE ATHLETE?

QUADRATIC FORMULA APPLICATIONS

• A PILOT OF A HELICOPTER PLANS TO RELEASE A BUCKET OF WATER ON A FOREST FIRE. THE HEIGHT Y IN FEET OF THE WATER T SECONDS AFTER ITS RELEASE IS MODELED BY Y = –16T2 – 2T + 500. THE HORIZONTAL DISTANCE X IN FEET BETWEEN THE WATER AND ITS POINT OF RELEASE IS MODELED BY X = 91T.

• THE PILOT’S ALTITUDE DECREASES, WHICH CHANGES THE FUNCTION DESCRIBING THE WATER’S HEIGHT TOY = –16T2 –2T + 400. TO THE NEAREST FOOT, AT WHAT HORIZONTAL DISTANCE FROM THE TARGET SHOULD THE PILOT BEGIN RELEASING THE WATER?

HOMEWORK

• NO HOMEWORK

CLOSURE

• TODAY WE LEARNED ABOUT THE QUADRATIC FORMULA AND ITS PROPERTIES.

• NEXT CLASS WE ARE GOING TO CONTINUE WITH THE QUADRATIC FORMULA