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:
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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!
Unit 2: Quadratics Chapter 4 – 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 = _______
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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?
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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?
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Chapter 4 – Quadratic Equations
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
Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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?
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
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
Unit 2: Quadratics Chapter 4 – Quadratic Equations
h) 𝑥2 + 6𝑥 = 1
i) 3𝑦2 = 8𝑦 − 3
Chapter 4 – Quadratic Equations
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!
Chapter 4 – Quadratic Equations
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?
Chapter 4 – Quadratic Equations
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?
How many solutions/roots
does this equation have?
3. 𝑥2 + 4𝑥 + 4 = 0 What does 𝑏2 − 4𝑎𝑐 equal?
How many solutions/roots
does this equation have?
Chapter 4 – Quadratic Equations
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
Chapter 4 – Quadratic Equations
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.
Chapter 4 – Quadratic Equations
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?