quadratic equations prepared by doron shahar. warm-up: page 15 a quadratic equation is an equation...

Chapter 1 Section 1.4Quadratic Equations

Prepared by Doron Shahar

Warm-up: page 15A quadratic equation is an equation that can be written in the form _____________ where a, b, and c are constants and a ≠ 0.

The zero product property says that if , then either ________ or ________.

4:for Solve

149:Factor

)3)(2(:FOIL

2

2

xx

xx

xx

02 cbxax

0A0AB

0B


FOIL and Factoring

)3)(2( xx 1492 xx

2x x3 x2 6

62 xx

)3)(2()32(2 xx

)2)(7( xx

1427 and 927

FirstOutsideInsideLast

FOIL Factor


1.4.1 Solve by Factoring01492 xx

0)2)(7( xx

2or 7 xx

Starting Equation

Factor

Solution

0)2(or 0)7( xxZero Product Property


1.4.2 Solve by Factoring

0982 xx

0)9)(1( xx

9or 1 xx

Starting Equation

Factor

Solution



Solve by Factoring6)3)(2( xx

0)3)(4( xx

3or 4 xx

Starting Equation

Factor

Solution


662 xxFOIL6 6

0122 xx


Intro to Completing the square42 x

42 x

2x

Starting Equation

Take square root of both sides of the equation

Solution

4xPlace ± on the right side of the equation


Intro to Completing the square6)1( 2x

61or 61 xx

Starting Equation

Insert ± on right side

Solution

6 1 x

6)1( 2 xTake square root

61xGroup like terms1 1


Goal of Completing the squareThe goal of completing the squares is to get a quadratic equation into the following form:

khx 2)(

khxkhx or

Starting Equation


Solution

kx h

khx 2)(Take square root

khx Group like terms

h h

6)1( 2 xeg.


1.4.5 Completing the square

03)1( 22

1 xStarting Equation3 3

3)1( 22

1 x

6)1( 2x

Add 3 to both sides

Multiply both sides by 2

Desired form

The goal of completing the squares is to get a quadratic equation into the following form: khx 2)(

32)1(2 22

1 x


Example: Completing the square

08)3( 22

1 xStarting Equation8 8

8)3( 22

1 x

16)3( 2x

Add 8 to both sides

Multiply both sides by 2

Desired form

82)3(2 22

1 x


Completing the square16)3( 2x

1or 7 xx

Equation from previous slide


Solution

4 3 x

16)3( 2 xTake square root

43xGroup like terms

3 3


1.4.2 General method of Completing the square

0982 xx

982 xx

1691682 xx

25)4( 2 x

Starting Equation

Add 9 to both sides

Add (−8/2)2=16 to both sides

Factor left side



823 2 xx

3

8

3

22 xx

9

1

3

8

9

1

3

22 xx

9

2523

1)( x

Starting Equation

Divide both sides by 3

Add ((−2/3)/2)2=1/9to both sides

Factor left side



02262 xx

2262 xx

922962 xx

13)3( 2 x

Starting Equation

Subtract 22 from both sides

Add (−6/2)2=9to both sides

Factor left side


No solutions in quadratic equation

Solving by factoring works only if the equation has a solution.

Completing the square always works, and can be used to determine whether a quadratic equation has a solution.

13)3( 2 x

13)3( 2 x

Try solving

Take square root

¡PROBLEMA! You CANNOT take the square root of a negative number. Therefore, the equation has no solution.


General method of Completing the square

02 cbxaxcbxax 2

22

2

22

a

b

a

c

a

b

a

b xx

22

22

)(a

b

a

c

a

bx

Starting Equation

Subtract c from both sides

Add ((b/a)/2)2=(b/2a)2

to both sides

Factor left side

Divide both sides by aa

c

a

b xx 2


Quadratic Formula

a

acbbx

2

42

22

22

)(a

b

a

c

a

bx If we solve for in the previous equation, ,we get an equation called the quadratic formula.

x

Quadratic Formula

Starting Equation 02 cbxaxThe quadratic formula gives us the solutions to every quadratic equation.

a

acbbx

2

42 Solution


Quadratic Formula Song

a

acbbx

2

42 Quadratic Formula

Please sing along.


Using the quadratic formulaStarting Equation 0169 2 xx

)9(2

)1)(9(4)6()6( 2 xSolution

9a 6b 02 cbxax1c

a

acbbx

2


Plug 9 in for a, 6 for b, and 1 for c in the quadratic formula.


Simplify your solution

)9(2

)1)(9(4)6()6( 2 xSimplify

18

06 x

18

06 x

18

6x

3

1xSolution


1.4.1 Using the quadratic formulaStarting Equation 01492 xx

)1(2

)14)(1(4)9()9( 2 xSolution

1a 9b 02 cbxax14c

a

acbbx

2


Plug 1 in for a, 9 for b, and 14 for c in the quadratic formula.

1



)1(2

)14)(1(4)9()9( 2 xSimplify

2

259 x

2

59 x

2

59or

2

59

xx

7or 2 xxSolution


1.4.3 Using the quadratic formulaStarting Equation 022 6 2 xx

)1(2

)22)(1(4)6()6( 2 xSolution

1a 6b 02 cbxax22c

a

acbbx

2


Plug 1 in for a, −6 for b, and 22 for c in the quadratic formula.

1 +( )



)1(2

)22)(1(4)6()6( 2 xSimplify

2

526 x

You cannot take the square root of a negative number. Therefore, there is no solution.

No Solution


DiscriminantEquation Discrimina

nt# of Solutions

0169 2 xx

01492 xx

022 62 xx

02 cbxax acb 42

0)1)(9(4)6( 2

025)14)(1(4)9( 2

052)22)(1(4)6( 2

04 if Solutions No

04 if Solutions 2

04 ifSolution 1

2

2

2

acb

acb

acb

Solution 1

Solutions 2

Solutions

No


CalculatorPut the Quadratic Formula program on your

calculator. Instructions are in the back of the class

notes.ORYou can come in to office hours to have me

load the program onto your calculator.Warning! The calculator will not always give you exact answers.


Simplifying expressions withbaab xxx 2 then ,0 If

18

25 25 5

29 29 23

20 54 54 52

2

206

2

526 2

52

2

6 53


Quadratic equations with DecimalsIf a quadratic equation has decimals, it is easiest to simply use

the quadratic formula. If you want, you can multiply both sides of the equation by a power of 10 (i.e., 10, 100, 1000, etc) to get rid of the decimals. This can make it easier to simplify the answer if you are evaluating the quadratic formula without a calculator.


Quadratic equations with FractionsIf a quadratic equation has fractions (and does not factor), it is

often easy to simply use the quadratic formula. If you want, you can multiply both sides of the equation by the least common denominator to get rid of the fractions. This can make it easier to simplify the answer.

If a quadratic equation has fractions (and factors), it is often easier to factor after having gotten rid of the fractions.


Variables in the denominatorsIf an equation has variables in the denominator, it is NOT a

quadratic equation. Such equations, however, can lead to linear equations.

We treat such equations like those with fractions. That is, we multiply both sides of the equation by a common denominator to get rid of the variables in the dominators. Ideally, we should multiply by the least common denominator.

Example: If our problem has B +16 in the denominator of one term,

and B in the denominator of another term, we multiply both sides of the equation by B(B+16). After the multiplication, the terms will have no variables in the denominators.


Variables in the denominators1

15

16

15

BB

1)16(15

16

15)16(

BB

BBBB

)16()16(1515 BBBB

Starting Equation

Multiply both sides by B(B+16)

)16(15

)16(16

15)16(

BB

BBB

BBB

The problem is now in a form you can solve.

Distribute the B(B+16)


Systems of equationsThere are two methods for solving systems of equations: Substitution and Elimination. Both work by combining the equations into a single equation with one variable. And sometimes the resulting equation leads to a quadratic equation.

We will only review substitution, because elimination is not a common method when working with systems of equations that lead to quadratic equations.


Substitution16BJ

11515

BJ

115

16

15

BB

Starting Equations

10or 24 BB

61610

or 401624

J

J

Substitute B+16 for J in the second equation

Solution for B

Plug in 24 and −10 for B in the first equation to get the solution for J


quadratic equations prepared by doron shahar. warm-up: page 15 a quadratic equation is an equation...

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