Math 20-1 Chapter 4 -...

Unit 2: Quadratics Chapter 4 – Quadratic Equations

Math 20-1

Chapter 4 Quadratic Equations

General Outcome: Develop algebraic and graphical reasoning through the study of relations.

Specific Outcomes: RF1. Factor polynomial expressions of the form:

• 𝑎𝑥2 + 𝑏𝑥 + 𝑐, a ≠ 0

• 𝑎2𝑥2 − 𝑏2𝑦2 , a ≠ 0, b ≠ 0

• 𝑎(𝑓(𝑥))2 + 𝑏(𝑓(𝑥)) + 𝑐, a ≠ 0

• 𝑎2(𝑓(𝑥))2 − 𝑏2(𝑔(𝑦))2 , a ≠ 0, b ≠ 0

where a, b and c are rational numbers. [CN, ME, R]

RF4. Analyze quadratic functions of the form y = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 to identify characteristics of the corresponding graph, including: • vertex • domain and range • direction of opening • axis of symmetry • x- and y-intercepts and to solve problems. [CN, PS, R, T, V] [ICT: C6–4.1, C6–4.3]

RF5. Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] [ICT: C6–4.1]

Mark Assignments

4.1: Page 215– 217 # 1 – 5, 9, 11, 17, 18

4.2: Page 229 – 233 # 1-2(ac), 3-4(bd), 5-7, 9-10(abdf), 30

4.3: Page 240 – 243 # 2-7(acf), 18.

4.4: Page 254 – 257 # (3, 4, 5, 7, 21-23) (1 & 2)

Quiz 4 Date:

Chapter 4 Test Date:


4.1 – Graphical Solutions of Quadratic Equations

Getting Started:

Solve this linear equation:

4𝑥 − 10 = 4 − 3𝑥

QUADRATIC FUNCTION –

A Quadratic Function is a function that can be written in the form:

𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 ∈ 𝑅 𝑎𝑛𝑑 𝑎 ≠ 0

or in equation form as:

𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 ∈ 𝑅 𝑎𝑛𝑑 𝑎 ≠ 0

QUADRATIC EQUATION –

A Quadratic Equation is an equation that can be written in the form:

𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 ∈ 𝑅 𝑎𝑛𝑑 𝑎 ≠ 0

SIMPLY PUT IT IS A QUADRATIC FUNCTION THAT IS EQUAL TO A SPECIFIC VALUE!!

In this chapter, we are going to learn methods of solving quadratic equations!


YOU TRY …

Given the equation 2𝑥2 + 12𝑥 + 16 = 0, what could you do to find the value(s) for

x that solve this equation??

(HINT: Where does 2𝑥2 + 12𝑥 + 16 equal zero??)

NOTE: The solutions to a quadratic equation are called the roots of the equation!

The graph of 𝑓(𝑥) = 𝑥2 + 4𝑥 + 3 is

shown. Fill in the blanks!

The GRAPH of

𝑓(𝑥) = 𝑥2 + 4𝑥 + 3

has x-intercepts

x = _______ and

x = _______.

The FUNCTION

𝑓(𝑥) = 𝑥2 + 4𝑥 + 3

has zeros

______ and ______ .

The EQUATION 𝑥2 + 4𝑥 + 3 =

0

has roots (or solutions)

x = _______ and

x = _______


How many x-intercepts can the graph of a quadratic function have?

Since these x-intercepts (zeros) represent the solutions to the corresponding

quadratic equation, how many solutions (roots) can a quadratic equation have?

Example 1

The flight path of a water skier jumper off of a ramp is modelled by the equation:

h(t) = 5.0 + 24.46t - 4.9t2

…where h(t) is the height of the jumper in meters and t is the time in seconds.

How long was he in the air for? Solve using your calculator.

When is the water-skier 15 m in the air?


Example 2

EXAMPLE 3: SOLVING QUADRATIC EQUATIONS GRAPHICALLY

Determine the roots of each quadratic equation? If necessary, round answers to

the nearest tenth.

a) 𝑥2 + 6𝑥 + 3 = 0

b) −𝑥2 = 2𝑥 + 1


c) 2𝑥2 + 22 = 12𝑥

EXAMPLE 4: CHECKING THE SOLUTIONS TO A QUADRATIC EQUATION

Solve each equation. Check your solutions.

Note: Check the solution(s) by substituting them into the ORIGINAL equation.

a) 2𝑥2 + 5𝑥 − 3 = 0

b) 𝑥2 − 6𝑥 = −9


EXAMPLE 5: SOLVE A PROBLEM INVOLVING QUADRATIC EQUATIONS

When a basketball is thrown, its height can be modeled by the function ℎ(𝑡) =

−4.9𝑡2 + 15𝑡 + 1, where h is the height of the ball, in metres, and t is time, in

seconds.

a) What does the equation 0 = −4.9𝑡2 + 15𝑡 + 1 represent in this situation?

b) How long was the basketball in the air?


4.2 – Factoring Quadratic Equations

Review:

Factor each of the following quadratic expressions:

1. 𝑥2 − 9𝑥 + 18

2. 𝑡2 + 𝑡 − 56

3. 5𝑎2 − 5𝑎 − 30

4. 3𝑛2 − 11𝑛 − 4

5. 2𝑥2 − 17𝑥 + 30

6. 1

4𝑥2 +

1

2𝑥 − 6

7. 0.5𝑧2 − 5.4𝑧 + 4

8. 9𝑥2 − 0.25𝑦2

9. 1

4𝑑2 −

1

49𝑓2

10. 8𝑎2 − 18𝑏2


Extending What You Know:

Factor each of the following quadratic expressions:

1. 9(𝑥 − 1)2 − 100𝑦2

2. 4(𝑥 + 3)2 + 8(𝑥 + 3) − 5

3. −4(𝑠 − 2)2 + 9(2𝑡 + 1)2

4. −2(𝑥 + 2)2 + 12(𝑥 + 2) + 9

KEY IDEAS:

Last class we learned one strategy for solving a quadratic is graphing. Another strategy is to use

FACTORING!!

The zero product property states that if the product of two real numbers is zero, then one or

both of the numbers must equal zero.

ie: This means if 𝑥𝑦 = 0, then …

This principle applies to quadratic equations.

ex: 𝑥2 + 2𝑥 − 24 = 0

(𝑥 + 6)(𝑥 − 4) = 0


EXAMPLE 1: SOLVING FACTORED EQUATIONS

Solve each equation. Note: They are already factored for you!!

a) (𝑥 − 5)(𝑥 + 9) = 0

b) (2𝑥 + 9)(𝑥 − 4) = 0

c) (𝑥 +3

4) (𝑥 −

11

2) = 0

d) 𝑥(3𝑥 − 14) = 0

Chapter 4 – Quadratic Equations

EXAMPLE 2: SOLVING QUADRATIC EQUATIONS BY FACTORING

Determine the roots of each quadratic equation. Verify/check your answers.

Note: Check the solution(s) by substituting them into the ORIGINAL equation.

d) 𝑥2 + 4𝑥 − 21 = 0

e) 𝑥2 = 10𝑥 − 16

f) 2𝑥2 = 9𝑥 + 5


EXAMPLE 3: SOLVING MORE QUADRATIC EQUATIONS BY FACTORING

Solve each equation.

c) 2𝑥2 + 18 = 12𝑥

d) 8𝑥2 − 18 = 0

e) 2𝑥2 = 4𝑥

f) (2𝑥 − 3)(𝑥 + 1) = 3


g) 2𝑥(𝑥 − 6) + 3𝑥 = 2𝑥 − 9

4.3 – Solving Quadratic Equations by Completing the Square

KEY IDEAS:

We can solve quadratic equations in vertex form 𝑦 = 𝑎(𝑥 − 𝑝)2 + 𝑞 by isolating

the squared term and taking the square root of both sides. THE SQUARE ROOT

OF A NUMBER CAN BE POSITIVE OR NEGATIVE SO THERE ARE TWO POSSIBLE

SOLUTIONS TO THESE EQUATIONS!!

EXAMPLE 1: TAKING SQUARE ROOTS TO SOLVE QUADRATIC EQUATIONS

Solve each equation. Verify your solutions.

e) 𝑥2 = 36

f) (𝑥 − 4)2 − 64 = 0


EXAMPLE 2: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE

Solve each quadratic equation by completing the square. Express your answers as

exact roots.

g) 𝑎2 + 18𝑎 + 32 = 0

h) 𝑡2 − 8𝑡 − 4 = 0

i) 4𝑠2 + 48𝑠 + 141 = 0


EXAMPLE 3: SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE

Determine the roots of −2𝑥2 − 3𝑥 + 7 = 0 to the nearest hundredth. Then, use

technology to verify your answers.

Why is it sometime more useful to complete the square then it is to solve an

equation graphically or by factoring?


4.4 – The Quadratic Formula

Developing the Quadratic Formula:

1. Complete the square to solve the following quadratic equation: 2𝑥2 + 7𝑥 + 1 = 0

2. Complete the square using the general quadratic equation in standard form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0


KEY IDEAS:

The Quadratic Formula:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

The quadratic formula can be used to solve any quadratic equation in general

form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0.

EXAMPLE 1: USING THE QUADRATIC FORMULA TO SOLVE EQUATIONS

Use the quadratic equation to solve each equation. Express your answers as

exact answers and to the nearest hundredth.

g) 9𝑥2 + 12𝑥 + 4 = 0


h) 𝑥2 + 6𝑥 = 1

i) 3𝑦2 = 8𝑦 − 3


EXAMPLE 2: SELECT A STRATEGY TO SOLVE A QUADRATIC EQUATION

Solve 6𝑥2 − 14𝑥 + 8 = 0 using each of the following methods:

GRAPHING:

FACTORING:

COMPLETING THE SQUARE:

THE QUADRATIC FORMULA:

Which of the above methods gives exact answers?

When deciding how to solve a quadratic:

1. Try factoring first!! (It’s the quickest method and gives an exact answer!) 2. If factoring is not possible, complete the square or use the quadratic

formula. (Either one will give you an exact answer – YOUR CHOICE!) 3. Use graphing as a way of checking all your answers!


4.4 – Part B – The Discriminant

KEY IDEAS:

The Quadratic Formula:

𝑥 =−𝑏 ± √𝑏2 − 4𝑎𝑐

2𝑎

The quadratic formula can be used to solve any quadratic equation in

general form: 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0, 𝑎 ≠ 0.

The Discriminant is the expression under the radical sign in the

quadratic formula:

𝑏2 − 4𝑎𝑐

The discriminant can be useful is providing us some key information about the

number of solutions or the nature of the roots of a quadratic equation. How?


Try solving each quadratic equation using the quadratic formula …

1. 𝑥2 − 5𝑥 + 4 = 0 What does 𝑏2 − 4𝑎𝑐 equal?

How many solutions/roots

does this equation have?

2. 8𝑥2 + 𝑥 + 17 = 0 What does 𝑏2 − 4𝑎𝑐 equal?



3. 𝑥2 + 4𝑥 + 4 = 0 What does 𝑏2 − 4𝑎𝑐 equal?




Value of the Discriminant

Nature of the Roots Graph of the Function

𝑏2 − 4𝑎𝑐 > 0

𝑏2 − 4𝑎𝑐 < 0

𝑏2 − 4𝑎𝑐 = 0

EXAMPLE: USING THE DISCRIMINANT TO DETERMINE THE NATURE OF THE

ROOTS

Use the discriminate to determine the nature of the roots. Check by graphing.

a) −2𝑥2 + 3𝑥 + 8 = 0

b) 3𝑥2 + 9 − 5𝑥 = 0

c) 1

4𝑛2 + 9 = 3𝑛


4.2 – 4.4 – Quadratic Formula Problem Solving

EXAMPLE 1:

Dock jumping is an exciting event in which dogs compete for the longest jumping

distance from a dock into a body of water. The path of a terrier on a particular

jump can be approximated by the quadratic function ℎ(𝑑) = −3

10𝑑2 +

11

10𝑑 + 2,

where h is the height above the water and d is the horizontal distance the dog

travels from the base of the dock. Both measurements are in feet.

Determine the horizontal distance of the jump.

EXAMPLE 2:

The length of a field is 10 m less than twice the width. The area is 6 600 m2.

Determine the dimensions of the field.


EXAMPLE 3:

A 42” widescreen television has a width that is 16 inches more than the height of

the screen. Determine the dimensions of the screen to the nearest tenth of an

inch.

EXAMPLE 4:

Leah wants to frame a painting measuring 50 cm by 60 cm. Before framing, she

places a mat so that a uniform strip of the mat shows on all sides of the painting.

The area of the mat is twice the area of the painting.

a) How wide is the strip of exposed mat (showing on all sides of the painting), to the nearest tenth of a centimetre?

b) What are the dimensions of the painting (and mat) to be framed?

Math 20-1 Chapter 4 -...

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