Transcript Lesson 4

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 10-4
Solving Quadratic Equations by
Using the Quadratic Formula
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Objectives
• Solve quadratic equations by using the
Quadratic formula
• Use the discriminant to determine the number
of solutions for a quadratic equation
Vocabulary
• Quadratic formula –
• Discriminant –
Working Backwards
• Start with the answer
• “Undo” the operation that got you to the
answer
• Keep “undoing” until you get back to the
beginning
Example 1
Use two methods to solve
Method 1
Factoring
Original equation
Factor
or
Zero Product Property
Solve for x.
Example 1 cont
Method 2
Quadratic Formula
For this equation,
Quadratic Formula
Multiply.
Example 1 cont
Add.
Simplify.
or
Answer: The solution set is {–5, 7}.
Example 2
Solve
by using the Quadratic Formula.
Round to the nearest tenth if necessary.
Step 1
Rewrite the equation in standard form.
Original equation
Subtract 4 from each side
Simplify.
Example 2 cont
Step 2
Apply the Quadratic Formula.
Quadratic Formula
a = 15, b = -8 and c = -4
Multiply, then Add.
or
Example 2 cont
Check the solutions by using the CALC menu on a
graphing calculator to determine the zeros of the related
quadratic function.
Answer: The approximate solution set is {–0.3, 0.8}.
Example 3
Space Travel Two possible future destinations of
astronauts are the planet Mars and a moon of the planet
Jupiter, Europa. The gravitational acceleration on Mars
is about 3.7 meters per second squared. On Europa, it
is only 1.3 meters per second squared. Using the
information and equation from Example 3 on page 548
in your textbook, find how much longer baseballs thrown
on Mars and on Europa will stay above the ground than
a similarly thrown baseball on Earth.
In order to find when the ball hits the ground, you must
find when H = 0. Write two equations to represent the
situation on Mars and on Europa.
Example 3 cont
Baseball Thrown on Mars
Baseball Thrown on Europa
These equations cannot be factored, and completing the
square would involve a lot of computation.
Example 3 cont
To find accurate solutions, use the Quadratic Formula.
Since a negative number is not reasonable, use the positive solutions.
Answer: A ball thrown on Mars will stay aloft 5.6 – 2.2 or
about 3.4 seconds longer than the ball thrown on Earth.
The ball thrown on Europa will stay aloft 15.6 – 2.2 or
about 13.4 seconds longer than the ball thrown on Earth.
Example 4a
State the value of the discriminant for
Then determine the number of real roots of the equation.
.
and
Simplify.
Answer: The discriminant is –220. Since the discriminant
is negative, the equation has no real roots.
Example 4b
State the value of the discriminant for
.
Then determine the number of real roots of the equation.
Step 1
Rewrite the equation in standard form.
Original equation
Add 144 to each side
Simplify.
Step 2
Find the discriminant.
a = 1, b = 24 and c = 144
Simplify.
Answer: The discriminant is 0. Since the discriminant is 0,
the equation has one real root.
Example 4c
State the value of the discriminant for
.
Then determine the number of real roots of the equation.
Step 1
Rewrite the equation in standard form.
Original equation
Subtract 12 from each side
Simplify.
Step 2
Find the discriminant.
a = 3, b =10 and c = -12
Simplify
Answer: The discriminant is 244. Since the discriminant
is positive, the equation has two real roots.
Summary & Homework
• Summary:
– The solutions of a quadratic equation in the form
ax2 + bx + c = 0, where a ≠ 0, are given by the
Quadratic Formula:
-b ± √b² - 4ac
x = ----------------------2a
• Homework:
– pg