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Unit III Quadratic Equations 1

Section 7.1 – Solving Quadratic Equations by Graphing

Investigating Solutions to Quadratic Equations

Example:

A missile fired from ground level is modeled by the quadratic function

h(t) = –20t2 + 120t, where h(t) represents height in meters and t is

the time in seconds.

(a) Indicate the positions along the quadratic trajectory above, where the rocket

attains a height of 0?

(b) For the function above, h(t) = –20t2 + 120t which variable is replaced

by the height 0 for those positions?

(c) Write the resulting quadratic when the variable mentioned above is

replaced by 0.

t

h(t)

Goal:

Solving Quadratic Equations by Graphing

Unit III Quadratic Equations 2

(d) Use graphing software (https://www.desmos.com/) to determine the times

below where the rocket attains a height of 0.

Remember:

Zeros of a Quadratic Function

What is a Quadratic Equation?

A second degree polynomial equation.

Standard Form of a quadratic Equation ax2 + bx + c = 0.

t

h(t)

x

y

Zeros of a Quadratic Function

●The zero(s) of a quadratic function represent

the position(s) where the height is _________

●The zero(s) of a quadratic function

are also referred to as the _________________

Solutions to Quadratic Equation

●Solutions to the quadratic equation

–20t2 + 120t = 0 are the ________________

of the quadratic function

●times where height is 0 are:

t1 = _____ and t2 = ______

h(t) = –20t2 + 120t

https://www.desmos.com/

Unit III Quadratic Equations 3

(e) Use the graph below to determine the times where the rocket attained a

height of 160 m.

t1 = _____ t2 = _____

(f) Use the quadratic function h(t) = –20t2 + 120t to write the quadratic

equation that represents the height at 160 m.

t1 2 3 4 5 6 7 8

h(t)

20 40 60 80

100 120 140 160 180 200

h(t) = –2t2 + 120t

How can we algebraically attain the time(s) to solve a quadratic

equation ax2 + bx + c = 0 such as: ______________________ ?

Unit III Quadratic Equations 4

Section 7.2 – Solving Quadratic Equations by Factoring

(I) Solving Quadratic Equations by Factoring

Example:

A missile fired from ground level is modeled by the quadratic function

h(t) = –20t2 + 120t, where h(t) represents height in meters and t is

the time in seconds.

t

h(t)

20 40 60 80

100 120 140 160 180 200

At a height of 160 m the quadratic function h(t) = –20t2 + 120t

develops into a quadratic equation:

160 = –20t2 + 120t

We can determine the two times where the missile attains a

height of 160 m by FACTORING the quadratic equation

160 = –20t2 + 120t

and using the zero product property to isolate and solve for t.

Goal:

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 5

Solve for t by factoring:

160 = –20t2 + 120t

(II) Review of Factoring

(A) Factoring by removing a common factor.

Example: Solve by factoring.

(i) –20x2 + 120x = 0 (ii) 12x2 = –8x

Express in the form ax2 + bx + c = 0

Remove the greatest common factor.

Factor the remaining trinomial.

To isolate t, apply the Zero Product Property

If the product of two real numbers is zero ( a • b = 0)

then one or both must be zero.

In other words: a = 0 and b = 0

Unit III Quadratic Equations 6

(II) Review of Factoring

(B) Solving quadratic equations by the difference of two squares.

Perfect Square Numbers or Factors to remember:

22 = 4

32 = 9

42 = 16

52 = 25

62 = 36

72 = 49

82 = 64

92 = 81

102 = 100

112 = 121

122 = 144

132 = 169

142 = 196

152 = 225

Example: Solve by factoring.

(i) 81x2 = 49 (ii) 121 – 4x2 = 0

Perfect Square Numbers or Factors

REMEMBER: The difference of two squares factors by pattern

a2 – b2 = (a – b)(a + b)

Unit III Quadratic Equations 7

(II) Review of Factoring

(C) Solving quadratic equations of the form x2 + bx + c = 0.

Example: Solve by factoring.

(i) x2 + 3x = 18 (ii) –2p2 – 20p + 48 = 0

(D) Solving quadratic equations of the form ax2 + bx + c = 0.

Example: Solve by factoring.

(i) 6x2 – 11x = 10 (ii) 8x2 = 22x – 15

P.411 #1 – #4

Unit III Quadratic Equations 8

Section 7.2 – Solving Quadratic Equations by Factoring

(Day 2)

(I) Determining Roots of a Quadratic Equation ax2 + bx + c = 0

●The roots of a quadratic equation represent the TWO

x values that are solutions to ax2 + bx + c = 0.

Example: Determine the ROOTS for:

(a) x(3x + 7) = 6

(b) –3.2x2 + 6.4x = –25.6 (c) − 1

3 𝑥2 = −𝑥 + 6

•Express each quadratic equation in

standard form ax2 + bx + c = 0 and factor.

•Set each factor equal to zero and solve

each linear equation.

Goal:

Solving Quadratic Equations by Factoring

Unit III Quadratic Equations 9

(VI) Applications of Quadratic Equation ax2 + bx + c = 0

Example:

The path of a missile shot into the air from a ship is modeled by the

quadratic function y= –4.9x2 + 39.2x + 44.1 where y represents the

height in meters and x is time in seconds.

Algebraically determine:

(a) How long will it take for the missile to hit the ocean?

(b) How long will it take for the missile to attain a height of 122.5 metres?

x

y Quadratic Path is modeled by:

y = –4.9x2 + 39.2x + 44.1

http://www.google.ca/imgres?q=war+ships&hl=en&safe=active&biw=1024&bih=568&gbv=2&tbm=isch&tbnid=hH0m3J9Vc7hPTM:&imgrefurl=http://captainsblog-ryley.blogspot.com/&docid=QbaRKJtLzQ6H2M&imgurl=http://2.bp.blogspot.com/-sqghzeBwYrA/TZAzwrLdrfI/AAAAAAAAABI/ExOJd3W6A3U/s1600/Battleship_Bismarck_Firing_A_Salvo.jpg&w=1024&h=768&ei=QiG0Tq2KMsr40gGM4Nm0BA&zoom=1&iact=rc&dur=0&sig=116108977142721622264&page=21&tbnh=162&tbnw=209&start=170&ndsp=8&ved=1t:429,r:7,s:170&tx=88&ty=82

Unit III Quadratic Equations 10

Height

time

10

10

20

20

30

30

40

40

50

50

Example:

An osprey dives toward the water to catch a salmon. Its height above the water, in

metres, t seconds after it begins its dive, is approximated by 2( ) 5 30 45h t t t .

Algebraically determine the time it takes for the osprey to reach a return height of

20 m.

Example:

A travel agency has 16 people signed up for a trip. The revenue for the trip is

modeled by the function

R(x) = –100x2 + 800x + 38400

where x represents the number of additional people to sign up. How many

additional people must sign up for the revenue to reach $40000?

P.411 – 413 #6, #8, #13, #14

http://www.google.ca/imgres?q=osprey&start=133&um=1&hl=en&safe=active&sa=N&biw=1024&bih=498&tbm=isch&tbnid=kt3t2QzoP7RNWM:&imgrefurl=http://www.stories-for-children.ca/fishing.php&docid=91eulttajWkO8M&imgurl=http://www.stories-for-children.ca/resources/osprey_fishing.jpg?timestamp=1334791817921&w=629&h=419&ei=PhOPULXfHsSG0QGOrYDgDg&zoom=1&iact=hc&vpx=706&vpy=184&dur=250&hovh=183&hovw=275&tx=208&ty=155&sig=107937703016156733068&page=11&tbnh=121&tbnw=182&ndsp=14&ved=1t:429,r:13,s:133,i:176 http://www.google.ca/imgres?q=osprey&start=120&um=1&hl=en&safe=active&sa=N&biw=1024&bih=498&tbm=isch&tbnid=Dho7Hj4voCWqDM:&imgrefurl=http://www.theballroomblog.com/2012/04/transcript-parrot-vs-raptor/osprey-in-flight-tom-munson/&docid=wRiNrTte5feTHM&imgurl=http://www.theballroomblog.com/wp-content/uploads/2012/04/osprey-in-flight-tom-munson.jpg&w=800&h=596&ei=PhOPULXfHsSG0QGOrYDgDg&zoom=1&iact=hc&vpx=340&vpy=70&dur=953&hovh=194&hovw=260&tx=166&ty=104&sig=107937703016156733068&page=10&tbnh=143&tbnw=257&ndsp=13&ved=1t:429,r:10,s:120,i:124

Unit III Quadratic Equations 11

Section 7.2 – Developing Quadratic Functions/Equations

(Day 3)

(I) Review of zeros and roots

Determining the zeros of a quadra