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Warm up
1) Graph y = x2 - 2x - 3
2) Use the graph to state the x-intercepts.
Note: the x-intercepts of the graph of y = x2 - 2x - 3
are the solutions of the equation 0 = x2 - 2x - 3
Unit 5 - Quadratic Equations
Quadratic Functions
Quadratic Equations
Solving Quadratic Equations by Graphing
Example 1: Solve by Graphing
- x2 + 3x = 2
Check your answers.
Example 2: Solve each quadratic equation by using a graphing calculator to graph each
corresponding quadratic function
1) 0 = x2 - 4x + 1
2) 9 = 3x2 + x
How many solutions can a quadratic equation have?
Practice
p. 275 #3 (graphing calculator)
#1, 2 (by hand)
You only have to do column 1 for all three questions.
You can use geogebra.org to check your answers.
Warm up
Solve for x
1) 1 - 3x = 8
2) 0 = 3(x + 1)2 – 12
We could also solve #2 by graphing
Solving Quadratic Equations by Factoring
Zero Product Property
Examples: Solve
1) (2x - 5)(x + 1) = 0
2) x2 + 7x + 12 = 0
Check your answers for #2
Examples: Solve (by factoring)
3) g2 - 8g = -16
4) 2x2 + 3 = 5x + 1
5) 3a2 - 5a = 0
6) 2b2 - 10b = 0
p. 282 #1, 2, and 4
Check your answer(s) for #6
7) 2x2 - 18 = 0
Warm up
Solve
1) 4x + 3 = 5
3) 6x2 + 13x = 5
2) (x - 1)(3x + 2) = 0
Solving by Factoring
1) Solve and check
2) Solve
3) Solve and check
p. 282 #5 – 10 (first column for all)
Warm up
Solve by factoring
1) x2 - 2x - 24 = 0
2) 30x2 + 2x - 4 = 0
Solving by Factoring
Word Problems
Example 1:
When the square or an integer is added to ten times the integer, the sum is zero. What is the
integer?
Example 2:
The product of two consecutive integers is 288. What are the numbers?
Example 3:
The length of a rectangle is double its width. The area of the rectangle is 162 m2. What are the
length and width?
Example 4:
One leg of a right triangle is 1 cm longer than the other leg. The length of the hypotenuse is 9 cm
longer than that of the shorter leg. Find the lengths of the three sides.
p. 283 #11,-14, 17
Warm up
Solve 4x2 + 12x + 9 = 0
Factored Form
vertex form
Example: y = x2 - 2x - 8
a) Find the x-intercepts.
b) Use the x-intercepts to find the vertex.
Example: y = 4x2 + 4x - 3
1) What is the y-intercept?
2) Determine the x-intercepts.
3) Determine the vertex.
Practice p. 287 #7, 8
Upcoming Quiz
- solve by factoring
- graphing using factored form
standard form
factored form
Solving using Quadratic Formula
Methods for solving 0 = ax2 + bx + c
Example 1: Solve 2x2 + 9x + 6 = 0
Example 3: Solve 0 = x2 - 4x + 7
Practice
p. 292 #1,2, 3, 5 (first column for all)
Example 2: Solve 4x2 - 12x = -9
Warm up
1) Solve by factoring
y2 - 4y - 21 = 0
2) Solve using the quadratic formula
m2 - m + 7 = 0
Application Problems
To solve ax2+bx+c=0, there are two options
1)
2)
Discriminant: the part under the root sign in the quadratic formula
-
if it is positive ...
-
if it is zero ...
-
if it is negative ...
Example:
Your deck currently measures 4 m by 3 m. You are going to extend it by increasing each
dimension by the same amount. If you want to double the area of the deck, how much
should you increase the length and width?
Practice
p. 293 #7, 11, 12, 13, 18
Warm up
Solve using the most efficient method
1) x2-3x-10=0
2) x2+4x-10=0
3) 2x2-3x+1=0
Solving Problems involving Quadratic Equations
There are essentially three types of problems if our relation is quadratic
1)
Given x, find the value of y
Sub x into the equation and simplify to find y
2)
Given y, find the value of x
By factoring or using quadratic formula
3)
Find the maximum or minimum value of y
By completing the square or using x-intercepts to find the vertex
Example 1)
The height of a toy rocket is modelled by the formula h = -4.9t2 + 60t + 3 where h is the height above
the ground in metres and t is the time in seconds after it is launched.
a)
What height is the rocket launched from?
b)
When does the rocket hit the ground?
c)
What is the maximum height of the rocket? What time does it reach that height?
d)
For how many seconds was the rocket above 100 m?
Example 2)
The “Quadratics Cup” is a coffee shop where mathies can buy delicious beverages with a mathematics
twist. They average 95 customers per day who buy a coffee for $2. The Mathnerds, who own the
shop, discover that if they decrease their price by $0.10 on the coffee, they will get 5 more customers
per day buying one.
a) Write an equation to model their revenue from medium coffee for one day.
b) What price should they set the coffee at to earn $176 from it per day? How many people would be
buying it at that price?
Practice
p. 294 #17
p. 305 #6, 7