Transcript Lesson 3

5-Minute Check on Chapter 2
Transparency 3-1
1. Evaluate 42 - |x - 7| if x = -3
2. Find 4.1  (-0.5)
Simplify each expression
4. (36d – 18) / (-9)
3. 8(-2c + 5) + 9c
5. A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops.
If one is chosen at random, what is the probability that it is
not green?
6.
Standardized Test Practice:
Which of the following is a true
statement
A
8/4 < 4/8
B
-4/8 < -8/4
C
-4/8 > -8/4
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D
-4/8 > 4/8
Lesson 10-3
Solving Quadratic Equations
by Completing the Square
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Objectives
• Solve quadratic equations by finding the
square root
• Solve quadratic equations by completing the
square
Vocabulary
• Completing the square –
Multiplication and Division PoE
Properties of Equality (PoE) are based on the concept
that as long as you do the same thing to both sides
of an equation, then you have not changed anything.
• Multiplication PoE
– For any numbers a, b, and c, if a = b, then ac = bc
– You can multiply both sides of an equation by the same
thing without changing the equation
• Division PoE
– For any numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c
– You can divide both sides of an equation by the same thing
without changing the equation
• Multiplication and division are reciprocal actions
Example 1
Solve
by taking the square root of each
side. Round to the nearest tenth if necessary.
is a perfect square
trinomial.
Take the square root of each side.
Simplify.
Definition of absolute value
Subtract 3 from each side.
Simplify.
Use a calculator to evaluate each value of x.
or
Answer: The solution set is {–5.2, –0.8}.
Example 2
Find the value of c that makes
Method 1
a perfect square.
Use algebra tiles.
is a perfect square.
Example 2 cont
Method 2
Complete the square.
Step 1
Find
Step 2
Square the result
of Step 1.
Step 3
Add the result of
Step 2 to
Answer:
Notice that
Example 3
Solve
Step 1
by completing the square.
Isolate the x2 and x terms.
Original equation
Subtract 5 from each side
Simplify.
Step 2
Complete the square and solve.
Since
,
add 81 to each side.
Factor
Example 3 cont
Factor
Take the square root of
each side.
Add 9 to each side.
Simplify.
or
Answer: The solution set is {1, 17}.
Example 4
Boating Suppose the rate
of flow of an 80-foot-wide
river is given by the equation
where r is the rate in miles per hour, and x is the distance
from the shore in feet. Joacquim does not want to paddle
his canoe against a current faster than 5 miles per hour. At
what distance from the river bank must he paddle in order to
avoid a current of 5 miles per hour?
Explore
You know the function that relates distance
from shore to the rate of the river current. You
want to know how far away from the river
bank he must paddle to avoid the current.
Example 4 cont
Plan
Solve
Find the distance when
the square to solve
Use completing
Equation for
the current
Divide each side
by –0.01.
Simplify.
Since ½(-80)² = 1600
add 1600 to each side.
Factor
Example 4 cont
Take the square root
of each side.
Add 40 to each side
Simplify.
Use a calculator to evaluate each value of x.
or
Examine
The solutions of the equation are about 7 ft
and about 73 ft. The solutions are distances
from one shore. Since the river is about 80 ft
wide,
Answer: He must stay within about 7 feet of either bank.
Summary & Homework
• Summary:
– Complete the square to make a quadratic function
a perfect square
– Use the following steps to complete the square of
x2 + bx:
• Step 1. Find ½ of b, the coefficient of x
• Step 2. Square the result of step 1
• Step 3. Add the result of step 2 to both sides of the
equation
• Homework:
– pg