Unit 2: Mid Checkpoint Apply, algebra homework help

Please complete the following questions. It is important that you show all work you did to solve the problems when you submit your work. This includes any calculations, diagrams, or graphs that helped you solve it.

1. OPEN ENDED Choose two integers. Then write an equation with those roots in standard form. How would the equation change if the signs of the two roots were switched?
2. OPEN ENDED Write two complex numbers with a product of 10.
3. OPEN ENDED Write a perfect square trinomial equation in which the linear coefficient is negative and the constant term is a fraction. Then solve the equation.
4. OPEN ENDED Graph a quadratic equation that has aa. positive discriminantb. negative discriminantc. zero discriminant
5. Writing in Math Use the information on page 273 to explain how to solve a quadratic equation using the Zero Product Property. Explain why you cannot solve x(x + 5) = 24 by solving x = 24 and x + 5 = 24.
6. Writing in Math Use the information on page 281 to explain how complex numbers are related to quadratic equations. Explain how the a and c must be related if the equation ax2+ c = 0 has complex solutions and give the solutions of the equation 2×2 + 2 = 0.

х
Х
Jeremy
Х
CC – Algebra 2
S Ask a new question – Stu x American High School
{-
www.intervisualtechnology.us/uploads/PDFs/ebooks/CC%20-%20Algebra%202/CC%20-%20Algebra%202/html5forpc.html?page=272&bbv=1&pcode=
:
5-3
Solving Quadratic Equations
by Factoring
EXTEND Graphing Calculator Lab
5-2
Modeling Using
Quadratic Functions
ACTIVITY
FALLING WATER Water drains from a hole made in a 2-liter bottle. The
table shows the level of the water y measured in centimeters from the
bottom of the bottle after x seconds. Find and graph a linear regression
equation and a quadratic regression equation. Determine which equation
is a better fit for the data.
Main Ideas
– Write quadratic
equations in
intercept for
• Solve quadratic
equations by
factoring
GET READY for the Lesson
The intercept form of a quadratic equation is
y = 2(X-P)(x-4). In the equation, p and
represent the x-intercepts of the graph
corresponding to the equation. The intercept
form of the equation shown in the graph is
y = 2(x – 1)( + 2). The x-intercepts of the
graph are 1 and 2. The standard form of the
equation is y=2×2 + 2x – 4
Time (s) o 20 40 80 120 140 160 186 200 220
Water level (cm) 42.6 40.7 38.9 322 358 343 333 32.3 315 308 304 30.1
New Vocabulary
interceptform
FOIL method
Intercept Form Changing a quadratic equation in intercept form to
standard form requires the use of the FOIL method. The FOIL method
uses the Distributive Property to multiply binomials.
10.250) sct:20 by 125.45] sc: 5
Step 1 Find a linear regression equation.
• Enter the times in Li and the water levels in L2. Then find a linear
regression equation. Graph a scatter plot and the equation.
KEYSTROKES: Review lists and finding and graphing a linear regression
equation on page 92.
Step 2 Find a quadratic regression equation.
. Find the quadratic regression equation. Then copy the equation
to the Y=list and graph.
KEYSTROKES: STAT 5 ENTER Y VARS 5 ENTER GRAPH
The graph of the linear regression equation appears to pass through
just two data points. However, the graph of the quadratic regression
equation fits the data very well.
KEY CONCEPT
FOIL Method for Multiplying Binomials
The product of two binomials is the sum of the products of F the first terms,
o the outer terms, I the inner terms, and the last terms.
To change y = 2(x – 1)(x + 2) to standard form, use the FOIL method to
find the product of (x – 1) and (x + 2), x2 + x -2, and then multiply
by 2. The standard form of the equation is y = 2x² + 2x – 4.
You have seen that a quadratic equation of the form (X-P)(x-4) = 0
has roots p and q. You can use this pattern to find a quadratic equation
for a given pair of roots.
TO, 260 sch: 20 by 125, 451 sc: 5
EXERCISES
For Exercises 1-4, use the graph of the braking distances for dry pavement.
1. Find and graph a linear regression equation and a
Average Braking Distance on
quadratic regression equation for the data.
Dry Pavement
Determine which equation is a better fit for the data.
2. Use the CALC menu with each regression equation
to estimate the braking distance at speeds of 100
and 150 miles per hour.
3. How do the estimates found in Exercise 2
compare?
4. How might choosing a regression equation that
does not fit the data well affect predictions made
20 30 45 55 60 65 80
by using the equation?
Speed (mph)
EXAMPLE Write an Equation Given Roots
Write a quadratic equation with and -5 as its roots. Write
the equation in the form ax? + bx + c = 0, where a, b, and
are integers.
(X-P)(x-4)= 0 Write the pattern.
(x – 2) [x – (-5) = 0 Replace p with and q with –5.
(x – 2) (x + 5) = 0 Simplity.
x2 + x – 0 Use FOIL
2×2 + 9x – 5 = 0 Multiply each side by 2 so that band care integers.
CHECK Your Progress
1. Write a quadratic equation with and 4 as its roots. Write the
equation in standard form.
Study Tip
Writing an
Equation
The pattern
(X-P)(x-4)=0
produces one equation
with rootsp and
In fact, there are an
infinite number of
equations that have
these same roots.
188
Source: Missouri Department of Revenue
252 Chapter 5 Quadratic Functions and Inequalities
Other Keystra esbraz 272-273 / 1100
American
Lesson 5-3 Solving Quadratic Equations by Factoring 253
High School
X
х
Jeremy
CC – Algebra 2
S Feed Site Studypool
American High School | X
→ C www.intervisualtechnology.us/uploads/PDFs/ebooks/CC%20-%20Algebra%202/CC%20-%20Algebra%202/html5forpc.html?page=272&bbv=1&pcode=
M
Reading Math
Imaginary Unitiis
usually written before
radical symbols to make
clear that it is not
under the radical
CHECK Your Progress Solve each equation. .
4A. 4×2 + 100 = 0
4B. x2 + 4 = 0
Operations with Complex Numbers Consider 5 + 2i. Since 5 is a real number
and 2i is a pure imaginary number, the terms are not like terms and cannot be
combined. This type of expression is called a complex number.
EXAMPLESquare Roots of Negative Numbers
Simplify.
a 18
b. V-125x
V-18 V-1.32.2 V-125×3 = -1.5.×4.5x
=V-1.732.2
V-1.752.V.V5x
= 1.3.2 or 3iV2
= 1.5.×2.5x or 5ix?V5x
CHECK Your Progress
2A. V-27
–
2B.V-216y*
The Commutative and Associative Properties of Multiplication hold true for
pure imaginary numbers.
EXAMPLE Products of Pure Imaginary Numbers
Simplify.
a.
a.-21.7
b.-10. V-15
-21.71 = -14:-
V-10.7-15 = iV10. i 15
= -14(-1) = -1
= 150
= -1.25.76
= -5/6
KEY CONCEPT
Complex Numbers
Words A complex number is any number that can be written in the form
a+bi, where a and b are real numbers and i is the imaginary unit.
a is called the real part, and b is called the imaginary part.
Examples 7+ 41 and 2 – 6 – 2 + (-6)
= 14
The Venn diagram shows the complex numbers,
• If b = 0, the complex number is a real number.
• Ifb #0, the complex number is imaginary.
• If a = 0, the complex number is a pure
imaginary number
Two complex numbers are equal if and only if
their real parts are equal and their imaginary
parts are equal. That is, a + bi = c + di if and
only if a = c and b = d.
b
Complex Numbers (a + bi)
Real Imaginary
Numbers Numbers
b=0 b0
Pure
Imaginar
Numbers
a=0
c”
Multiplying powers
= 1. (2)2 Power of a Power
= 1. (-1)2 = -1
= i. 1 ori (-1)22 – 1
CHECK Your Progress
3A. 31.41
38.-20. V-12
3C. 31
Reading Math
Complex Numbers
The form + Wis
sometimes called the
standard form of a
complex number
You can solve some quadratic equations by using the Square Root Property.
Square Root Property
Reading Math
Plus or Minus tvis
read plus or minus the
Square root of
KEY CONCEPT
For any real number n, if x? = n, then x = tvn.
EXAMPLE Equate Complex Numbers
Find the values of x and y that make the equation
2x – 3 + (y – 4)i = 3 + 2i true.
Set the real parts equal to each other and the imaginary parts equal to
each other.
2x – 3 = 3 Real parts
y – 42 Imaginary parts
2x = 6 Add to each side.
=
y = 6 Add 4 to each side
x = 3 Divide each side by 2
CHECK Your Progress
5. Find the values of x and y that make the equation
5x + 1 + (3 + 2y)i = 2x – 2 + (y – 6) i true.
EXAMPLE Equation with Pure Imaginary Solutions
Solve 3×2 + 48 = 0.
3x + 48 = 0 Original equation
3×2 = -48
Subtract 48 from each side.
x2 = -16 Divide each side by 3.
x= -16 Square Root Property
= +41 V-16 = V 16. V-1
260 Chapter 5 Quadratic Functions and Inequalities
To add or subtract complex numbers, combine like terms. That is, combine
the real parts and combine the imaginary parts.
Lesson 5-4 Complex Numbers 261