Quadratic Quadratic EquationsEquations

MTED 301MTED 301

May 7, 2008 May 7, 2008

Diane YumDiane Yum

Quadratic EquationQuadratic Equation

An equation that could be written as An equation that could be written as axax22+bx+c=0+bx+c=0

Quadratic EquationQuadratic Equation

An equation that could be written as An equation that could be written as axax22+bx+c=0+bx+c=0Standard formStandard form of a quadratic equation of a quadratic equationaxax22+bx+c=0+bx+c=0

Quadratic EquationQuadratic Equation

An equation that could be written as An equation that could be written as

axax22+bx+c=0+bx+c=0

Standard formStandard form of a quadratic equation of a quadratic equation

axax22+bx+c=0+bx+c=0

- The quadratic is on the left and 0 is - The quadratic is on the left and 0 is on the right. on the right.

- - Moreover, it is standard for the Moreover, it is standard for the leading coefficient leading coefficient AA to be to be positivepositive. .

3 different ways to solve a quadrati3 different ways to solve a quadratic equationc equation

3 different ways to solve a 3 different ways to solve a quadratic equationquadratic equation

Solve by Solve by FactoringFactoring

3 different ways to solve a quadrati3 different ways to solve a quadratic equationc equation

Solve by Solve by FactoringFactoringSolve by Solve by Completing the SquCompleting the Squ

areare

3 different ways to solve a quadrati3 different ways to solve a quadratic equationc equation

Solve by Solve by FactoringFactoringSolve by Solve by Completing the SquCompleting the Squ

areareSolve by Using “Solve by Using “Quadratic FoQuadratic Fo

rmularmula” ”

Solving by FactoringSolving by Factoring To solve a quadratic equation, To solve a quadratic equation,

①① Put all terms on one side of the equal Put all terms on one side of the equal sign, leaving zero on the other side sign, leaving zero on the other side ((Standard FormStandard Form) )

②② FactorFactor

③③ Set each factor equal to Set each factor equal to zerozero

④④ Solve each of these equationsSolve each of these equations

⑤⑤ CheckCheck by inserting your answer in the by inserting your answer in the original equation original equation

Example of factoringExample of factoring

Ex) Solve for y: yEx) Solve for y: y22 = -6y – 5 = -6y – 5

First, change the equation into the First, change the equation into the standard formstandard form: y: y22 + 6y + 5 = 0 + 6y + 5 = 0

FactoringFactoring, (y+5) (y+1) = 0, (y+5) (y+1) = 0

Y+5 = 0 or y+1 = 0Y+5 = 0 or y+1 = 0

Y = -5 or y = -1Y = -5 or y = -1

CheckCheck your answer your answer(-5)(-5)22 = -6(-5) – 5 or (-1) = -6(-5) – 5 or (-1)22 = -6(-1) -5 = -6(-1) -5

25 = 30 – 5 1 = 6 - 525 = 30 – 5 1 = 6 - 52525 = 25 1 = 1 = 25 1 = 1

You got it right You got it right

Solving by Completing the Solving by Completing the SquareSquare

Completing the SquareCompleting the Square: Finding : Finding something to add to a quadratic to something to add to a quadratic to make it a perfect make it a perfect

squaresquare Expression: Expression: (x+k)(x+k)22

Applying our formula for squaring a Applying our formula for squaring a

binomial, we get binomial, we get

(x+k)(x+k)22 = x = x22 + 2xk + k + 2xk + k22

So if you have an expression of the form So if you have an expression of the form

xx22+bx and you want to find something +bx and you want to find something to to

add to it to make it a perfect square, add to it to make it a perfect square, then you need to then you need to

Divide b by 2 to get k  Divide b by 2 to get k  Square k to get kSquare k to get k22

Ex) y2 – 9y Ex) y2 – 9y 2

929

4

81)

2

9( 2

22 )2

9(

4

819 yyy

Example of Completing the Example of Completing the SquareSquare

Ex) Ex) Solve Solve xx22 + 6 + 6xx – 7 = 0 by completing – 7 = 0 by completing the squarethe square

1)1) xx22 + 6x – 7 = 0 + 6x – 7 = 02)2) xx22 + 6x = 7 + 6x = 73)3) (6/2)(6/2)22 = 9 = 94)4) xx22 + 6x + 9 = 7 + 9 + 6x + 9 = 7 + 95)5) (x + 3 )(x + 3 )22 = 16 = 166)6) x + 3 = +4, -4x + 3 = +4, -47)7) x = -3 + 4 and -3 – 4x = -3 + 4 and -3 – 48)8) x = +1 and -7x = +1 and -7

CheckCheck your answer your answer

xx22 + 6 x – 7 = 0 + 6 x – 7 = 0

(1)(1)22 + 6(1) – 7 = 0 + 6(1) – 7 = 0 andand

1 + 6 – 7 = 01 + 6 – 7 = 0

You got it right again You got it right again

xx22 + 6x – 7 = + 6x – 7 = 00

(-7)(-7)22 + 6(-7) – 7 = + 6(-7) – 7 = 00

49 – 42 - 7 = 49 – 42 - 7 = 00

Solving by Quadratic Solving by Quadratic FormulaFormula

Quadratic Formula :Quadratic Formula :

Easy Steps to solve by quadratic formulaEasy Steps to solve by quadratic formula

1)1) Find a, b, and c in the standard formFind a, b, and c in the standard form2)2) Substitute numbers of a, b, and c in the Substitute numbers of a, b, and c in the

quadratic formulaquadratic formula3)3) Find the value of x Find the value of x

a

acbbx

2

42

Example of Quadratic Example of Quadratic FormulaFormula

Ex) Solve the equation of 2xEx) Solve the equation of 2x22 + 5x = + 5x = 10 by using a 10 by using a quadratic formulaquadratic formula

① ① Rewrite the equation into a standard form Rewrite the equation into a standard form 2x2x22 + 5x – 10 = 0 + 5x – 10 = 0

② ② Identify the values of a, b, and c Identify the values of a, b, and c a = 2, b = 5, c = -10a = 2, b = 5, c = -10

③③Substitute these values into the Quadratic Substitute these values into the Quadratic Formula Formula

SubstitutionSubstitution

You can substitute the x values into You can substitute the x values into the original equation to the original equation to checkcheck the the answer! answer!

4

1055

4

80255

)2(2

)10)(2(4)5(5 2

x

4

1055

4

1055 andx

HomeworkHomeworkDue : Next Class MeetingDue : Next Class Meeting

Solve each equation by Solve each equation by factoringfactoring, , completing the completing the squaresquare, or the , or the quadratic quadratic formulaformula..

Solve (x+1)(x-3) = 0Solve (x+1)(x-3) = 0 Solve xSolve x22 + x – 4 = 0 + x – 4 = 0 Solve xSolve x22 – 3x – 4 = 0 – 3x – 4 = 0 Solve 6xSolve 6x22 + 11x – 35 = 0 + 11x – 35 = 0 Solve xSolve x22 – 48 = 0 – 48 = 0