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Lecture 30Lecture 30 11
Solving Quadratic EquationsSolving Quadratic Equations
Two MethodsTwo Methods
Unit 4 Lecture 30Unit 4 Lecture 30Solving Quadratic EquationsSolving Quadratic Equations
Lecture 30Lecture 30 22
Solving Quadratic EquationsSolving Quadratic Equations
0 =0 = a axx22 ++ b bx +x + c c
Lecture 30Lecture 30 33
ObjectivesObjectives
• Solve a quadratic equation by factoringSolve a quadratic equation by factoring
• Solve a quadratic equation by using the Solve a quadratic equation by using the quadratic formulaquadratic formula
Lecture 30Lecture 30 44
Solving Quadratic EquationsSolving Quadratic Equations
Find values of x that make Find values of x that make the equation = 0the equation = 0
0 =0 = a axx22 ++ b bx +x + c c
Sometimes called the Sometimes called the zeroszeros or or rootsroots of the equation of the equation
In graphing they are called the In graphing they are called the x-interceptsx-intercepts
Lecture 30Lecture 30 55
Solving Quadratic EquationsSolving Quadratic Equations
•Factor the quadratic, if possibleFactor the quadratic, if possible
•Remember that if A * B = 0, then Remember that if A * B = 0, then either A = 0 or B = 0, or both = 0either A = 0 or B = 0, or both = 0•Use the quadratic formula, if the Use the quadratic formula, if the factors are not obviousfactors are not obvious
Lecture 30Lecture 30 66
Solve: Solve: xx2 2 + 5x + 6=0+ 5x + 6=0
Lecture 30Lecture 30 77
Solve: Solve: xx2 2 + 5x + 6=0+ 5x + 6=0
(x + 3)(x + 2)=0(x + 3)(x + 2)=0
Lecture 30Lecture 30 88
Solve: Solve: xx2 2 + 5x + 6=0+ 5x + 6=0
(x + 3)(x + 2)=0(x + 3)(x + 2)=0
x + 3 =0 or x + 2 = 0x + 3 =0 or x + 2 = 0
Lecture 30Lecture 30 99
Solve: Solve: xx2 2 + 5x + 6=0+ 5x + 6=0
(x + 3)(x + 2)=0(x + 3)(x + 2)=0
x + 3 =0 or x + 2 = 0x + 3 =0 or x + 2 = 0
x = -3x = -3 x = -2x = -2oror
Lecture 30Lecture 30 1010
Solve: Solve: xx2 2 - 12x + 35 =0- 12x + 35 =0
Lecture 30Lecture 30 1111
Solve: Solve: xx2 2 - 12x + 35 =0- 12x + 35 =0
(x - 7)(x - 5)=0(x - 7)(x - 5)=0
Lecture 30Lecture 30 1212
Solve: Solve: xx2 2 - 12x + 35 =0- 12x + 35 =0
(x - 7)(x - 5)=0(x - 7)(x - 5)=0
x - 7 =0 or x - 5 = 0x - 7 =0 or x - 5 = 0
Lecture 30Lecture 30 1313
Solve: Solve: xx2 2 - 12x + 35 =0- 12x + 35 =0
(x - 7)(x - 5)=0(x - 7)(x - 5)=0
x - 7 =0 or x - 5 = 0x - 7 =0 or x - 5 = 0
x = 7x = 7 oror
Lecture 30Lecture 30 1414
Solve: Solve: xx2 2 - 12x + 35 =0- 12x + 35 =0
(x - 7)(x - 5)=0(x - 7)(x - 5)=0
x - 7 =0 or x - 5 = 0x - 7 =0 or x - 5 = 0
x = 7x = 7 x = 5x = 5oror
Lecture 30Lecture 30 1515
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
Lecture 30Lecture 30 1616
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
(x + 6)(x - 1)=0(x + 6)(x - 1)=0
Lecture 30Lecture 30 1717
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
(x + 6)(x - 1)=0(x + 6)(x - 1)=0
x + 6 =0 or x - 1 = 0x + 6 =0 or x - 1 = 0
Lecture 30Lecture 30 1818
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
(x + 6)(x - 1)=0(x + 6)(x - 1)=0
x + 6 =0 or x - 1 = 0x + 6 =0 or x - 1 = 0
x = -6x = -6 oror
Lecture 30Lecture 30 1919
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
(x + 6)(x - 1)=0(x + 6)(x - 1)=0
x + 6 =0 or x - 1 = 0x + 6 =0 or x - 1 = 0
x = -6x = -6 x = 1x = 1oror
Lecture 30Lecture 30 2020
Solve: Solve: xx2 2 + 5x – 6+ 5x – 6 =0=0
(x + 6)(x - 1)=0(x + 6)(x - 1)=0
x + 6 =0 or x - 1 = 0x + 6 =0 or x - 1 = 0
x = -6x = -6 x = 1x = 1oror
Lecture 30Lecture 30 2121
Solve: Solve: xx2 2 - 5x - 6- 5x - 6 =0=0
Lecture 30Lecture 30 2222
Solve: Solve: xx2 2 - 5x - 6- 5x - 6 =0=0
(x - 6)(x + 1)=0(x - 6)(x + 1)=0
Lecture 30Lecture 30 2323
Solve: Solve: xx2 2 - 5x - 6- 5x - 6 =0=0
(x - 6)(x + 1)=0(x - 6)(x + 1)=0
x - 6 =0 or x + 1 = 0x - 6 =0 or x + 1 = 0
Lecture 30Lecture 30 2424
Solve: Solve: xx2 2 - 5x - 6- 5x - 6 =0=0
(x - 6)(x + 1)=0(x - 6)(x + 1)=0
x - 6 =0 or x + 1 = 0x - 6 =0 or x + 1 = 0
x = 6x = 6 oror
Lecture 30Lecture 30 2525
Solve: Solve: xx2 2 - 5x - 6- 5x - 6 =0=0
(x - 6)(x + 1)=0(x - 6)(x + 1)=0
x - 6 =0 or x + 1 = 0x - 6 =0 or x + 1 = 0
x = 6x = 6 x = -1x = -1oror
Lecture 30Lecture 30 2626
Solve: Solve: xx2 2 - 5x - 4- 5x - 4 =0=0
Lecture 30Lecture 30 2727
Solve: Solve: xx2 2 - 5x - 4- 5x - 4 =0=0
(x - )(x + )=0(x - )(x + )=0
Lecture 30Lecture 30 2828
Solve: Solve: xx2 2 - 5x - 4- 5x - 4 =0=0
(x - 4)(x + 1 )=0(x - 4)(x + 1 )=0
NO, NO, NO!NO, NO, NO!
Lecture 30Lecture 30 2929
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
Lecture 30Lecture 30 3030
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
Lecture 30Lecture 30 3131
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
Lecture 30Lecture 30 3232
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1 b = -5b = -5
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
Lecture 30Lecture 30 3333
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1 b = -5b = -5 c = -4c = -4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
Lecture 30Lecture 30 3434
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1 b = -5b = -5 c = -4c = -4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
2 (1 (4
2 )
5
1
))5
(
4x
Lecture 30Lecture 30 3535
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1 b = -5b = -5 c = -4c = -4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
2 (1 (4
2 )
5
1
))5
(
4x
5 25 16
2x
Lecture 30Lecture 30 3636
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
a = 1a = 1 b = -5b = -5 c = -4c = -4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
2 (1 (4
2 )
5
1
))5
(
4x
5 25 16
2x
Lecture 30Lecture 30 3737
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 = x0 = x2 2 - 5x - 4- 5x - 4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
5 41
2x
5 25 16
2x
Lecture 30Lecture 30 3838
To find the square root of a number, use 2nd key , then use key in row 6 column 1.
Use of the calculator to Use of the calculator to evaluate a square root.evaluate a square root.
is keyed in as is keyed in as 2nd2nd, ,, , 41 41, , ENTERENTER
41
6.4036.403
Lecture 30Lecture 30 3939
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 = x0 = x2 2 - 5x - 4- 5x - 4
0 =0 = a axx22 ++ b bx +x + c c2 4
2x
cb b a
a
5 41
2x
5 25 16
2x
5 6.4
2x
Lecture 30Lecture 30 4040
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
5 6.4 11.45.75 6.4
2 22
x
5 6.4
2x
Lecture 30Lecture 30 4141
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
5 6.4 11.45.7
5 6.4 2 22 5 6.4 1.4
.72 2
x
5 6.4
2x
Lecture 30Lecture 30 4242
Use the Quadratic Formula to solve:Use the Quadratic Formula to solve:
0 =0 = x x2 2 - 5x- 5x - 4- 4
5 6.4 11.45.7
5 6.4 2 22 5 6.4 1.4
.72 2
x
5 6.4
2x
Two solutions: x = 5.7 and x = -.7Two solutions: x = 5.7 and x = -.7
