Section 5'7 Applications of Quadratic Equations

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Section 5.7Applications of Quadratic Equations
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Area Example
w
48
l w 2

• The length of a room is 2 meters more than the
  width. The area of the floor is 48 square
  meters. Find the length and width of the room.

Can w -8?
Thus w 6
And l w 2 l 8
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Complicated Example

• The length of each side of a square is increased
  by 4 inches.
• The sum of the areas of the original square and
  the larger square is 106 square inches.
• What is the length of a side of the original
  square?

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The length of each side of a square is increased
by 4 inches.
x 4
x
Area x2
Area (x4)2
x2 8x 16
The sum of the areas of the original square and
the larger square is 106 square inches.
x2 x2 8x 16 106
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Algebra

• x2 x2 8x 16 106
• 2x2 8x - 90 0
• 2(x2 4x - 45) 0
• 2(x 9)(x - 5) 0
• x -9, 5
• Reject -9
• so x 5 and x 4 9

So the original square measures 5inches on a side.
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Consecutive Integers

• 1
• 2
• 3
• 4
• 5

• x
• x 1
• x 2
• x 3
• x 4

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Consecutive Even Integers

• 0
• 2
• 4
• 6
• 8

• x
• x 2
• x 4
• x 6
• x 8

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Consecutive Odd Integers

• 1
• 3
• 5
• 7
• 9

• x
• x 2
• x 4
• x 6
• x 8

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Product of Consecutive Even

• The product of two consecutive even integers is 4
  more than two times their sum. Find the integers.

x(x2)

4
2(x(x2))
x-2, 4
x2 2x 42(2x2))
-2, 0 is one pair
x2 2x 4 4x 4
x2 - 2x - 8 0
4, 6 is the other
(x 2)(x - 4) 0
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Consecutive Odd

• Find three consecutive odd integers such that the
  product of the smallest and the largest is 16
  more than the middle integer.

x(x4)

16
(x 2)
x2 4x 16 x 2
x x 2 x 4
3 5 7
x2 3x - 18 0
(x 6)(x - 3) 0
x - 6, 3
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Pythagorean Theorem
In a right triangle, the side opposite the right
angle is called the hypotenuse.
A triangle with a 90º angle is called a right
triangle.
The other two sides are called legs.
The small square indicates a 90º angle.
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Pythagorean Theorem

• If you square the legs of a right triangle and
  then add...
• The result will be the same as the square of the
  Hypotenuse.

c
c
a2 b2 c2
a
3
32 42 c2
9 16 c2
b
4
25 c2
5 c
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Pythagorean Theorem
52
32
9 16 25
42
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Right Triangle Example
c
x 3
a
x - 3

• The hypotenuse of a right triangle is 3 inches
  longer than the longer leg.
• The shorter leg is 3 inches shorter than the
  longer leg.
• Find the lengths of the sides of the triangle.

b
x
a2 b2 c2
(x - 3)2 x2 (x 3)2
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Right Triangle Example
x 3
x 3 15
x - 3 9
x - 3
(x - 3)2 x2 (x 3)2
x
x 12
x2 - 3x - 3x 9 x2 x2 3x 3x 9
x2 - 12x 0
9 inches 12 inches 15 inches
x(x - 12) 0
x 0, 12
Reject x 0
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What is the difference?

• (x 3)(x 2) 0

x-3, 2

• (x 3)(x - 2) 24

x2 x 6 24
x2 x 30 0
x-6, 5
(x 6)(x 5) 0
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Right Triangle Problem
c
15
a
x - 3

• The hypotenuse of a right triangle is 15 inches
  long.
• The shorter leg is 3 inches shorter than the
  longer leg.
• Find the perimeter of the triangle.

b
x
a2 b2 c2
(x - 3)2 x2 (15)2
Now for a bunch of algebra.
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(x - 3)2 x2 (15)2
(x - 3)2 (x - 3)(x - 3)
x2 - 3x - 3x 9
x2 - 6x 9 x2 225
x2 - 6x 9
2x2 - 6x 9 - 225 0
2x2 - 6x - 216 0
2(x2 - 3x - 108) 0
2(x - 12)(x 9) 0
x 12, -9
Now back to the problem.
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Right Triangle Problem
c
15
a
x - 3
9

• The hypotenuse of a right triangle is 15 inches
  long.
• The shorter leg is 3 inches shorter than the
  longer leg.
• Find the perimeter of the triangle.

12
x
b
a2 b2 c2
(x - 3)2 x2 (15)2
x 12, -9
Perimeter 91215
Perimeter 36
Reject -9
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Consecutive Evens

• The product of two consecutive even integers is
  168. Find the integers.

x(x2)

168
x2 2x 168
x2 2x - 168 0
-14, -12 is one pair
(x - 12)(x 14) 0
12, 14 is the other
x-14, 12
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Homework Section 5.7

• 1-31 odd