Quadratic Equations p' 110119 1'4

1
Quadratic Equations p. 110-119 (1.4)

• OBJECTVES
• Find x-intercepts by factoring
• Find x-intercepts by extracting square roots
• Find x-intercepts by completing the square
• Find x-intercepts using the quadratic formula

2
Quadratic equation in x an equation that can be
written in the general form ax2 bx c
0 where a, b, and c, are real numbers . A
quadratic equation is also known as a
second-degree polynomial equation in x, p. 110.
3

• Solve 2x2 19x 33 for x by factoring.Rf.
  p.33-37

1. Set equation equal to zero, called general
form.
2x2 19x 33 0
2. Express as a product of linear factors, called
factoring (p. 36)
2x2 19x 33 0
3
-11
( 2x ) ( x ) 0
4

• 3. Set each linear factor equal to zero and solve
  for x

( 2x 3 ) ( x 11 ) 0 2x 3 0 , x 11
0 x 3/2 , x 11
We used the Zero Factor Property p. 110 (p.
8) If ab 0 , then a 0 or b 0.
F O I L 2x2 22x 3x 33 0 2x2 19x 33
0
Check Rf. p.24-28 ( 2x 3 ) ( x 11 ) 0
5

• Solve 2x2 3x 5 0 for x by factoring.

5
1
( 2x ) ( x ) 0
2x 5 0 , x 1 0 2x 5 ,
x 1 x 5/2 , x 1 Try p. 120
7-20
6
Area using quadratics p. 121 109
The floor of a one-story building is 14 feet
longer than it is wide. The building has 1632
square feet of floor space.
w 14

• width w
• length w 14
• Area 1632

w
VERBAL MODEL
Length

Width
Area
ALGEBRAIC lw A MODEL
7

• ALGEBRAIC lw A
• MODEL
• ( w 14 ) w 1632
• w2 14w 1632
• w2 14w 1632 0
• ( w 34 ) ( w 48 ) 0
• w 34 0 , w 48 0
• w 34 , w -48
• Therefore, the width is 34 and the length is 48.

8
Extracting Square Roots p.111
The equation u2 d, where d gt 0, has exactly two
solutions and These solutions can also be
written as
9
p.120 24 Solve the equation x2 32 by
extracting the square roots.
EXACT ANSWER !!
List both the exact solution and the decimal
solution rounded to two decimal places.
or
10

• p.120 28 Solve the equation ( x 13 ) 2 25
  by extracting the square roots.

x -13 5 or x -13 - 5
x -8 or x -18 Try p. 120 21-34
11
Position equation- an equation that gives the
height of an object that is falling, p. 117. s
-16t2 v0t s0s height of the object (above
ground)v0 initial velocitys0 initial height
of objectt time

• The Empire State building is 1453 feet tall.
• Suppose an object falls from rest the position
  equation is
• s -16t2 (0)t 1453 or
• s -16t2 1453

12

• p. 120 116 Suppose King Kong falls from the top
  of the empire state building.
• a) Use the position equation to write a
  mathematical model for the height of King Kong.
• s -16t2 1453
• b) Find the height of King Kong after 4 seconds.
• s -16(4)2 1453
• s -16(16) 1453
• s -256 1453
• s 1197

13

• c) How long will it take before Kong hits the
  ground.
• s -16t2 1453
• 0 -16t2 1453
• 16t2 1453
• or approximately 10 seconds

14

• p. 120 38
• Solve the quadratic equation x2 8x 14 0 by
  completing the square.
• x2 8x 14
• x2 8x 14

• x2 8x 42 14 42
• x2 8x 16 2
• ( x 4 )( x 4 ) 2
• ( x 4 )2 2

15

• Consider again 2x2 3x 5 0 with solutions
  x 5/2 , x 1 . Solve the equation by
  completing the square.

2x2 3x 5 1(2x2 3x 5) 2
16
x 4/4 or x -10/4,
17
Completing the Square p. 111To complete the
square for the expression add , which is
the square of half the coefficient of x.
Consequently,Try p. 120 35-44

• The Quadratic Formula p. 112
• The solutions of a quadratic equation in the
  general form
• are given by the Quadratic Formula

18

• Consider again 2x2 3x 5 0 with solutions x
  5/2 , x 1 . Solve the equation by using
  the quadratic formula.
• 2x2 3x 5 0
• a 2, b 3 , c 5

x 4/4 or x -10/4 x 1 or x -5/2 Try p.
121 67-90
19

• The solutions of a quadratic equation (p. 115)
• can be classified as follows.
• If the discriminant b2 4ac is
• 1. positive, the equation has two distinct real
  solutions and its graph has two x-intercepts.
• 2. zero, the equations has one repeated real
  solution and its graph has one x-intercept.
• 3. negative, the equation has no real solution
  and its graph has no x-intercept.

20
Homework

• Work p. 120-125 1-50, 59-116, 121-158 alt.odd
• Read p. 126-130
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