2 11 homework answers

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
• All solvers
• Arithmetics
• Determinant
• Percentages
• Scientific Notation
• Inequalities

What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
• The inequalities section lets you solve an inequality or a system of inequalities for a single variable. You can also plot inequalities in two variables.
• The calculus section will carry out differentiation as well as definite and indefinite integration.
• The matrices section contains commands for the arithmetic manipulation of matrices.
• The graphs section contains commands for plotting equations and inequalities.
• The numbers section has a percentages command for explaining the most common types of percentage problems and a section for dealing with scientific notation.

Math Topics

More solvers.

• Add Fractions
• Simplify Fractions

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Want additional help with Algebra 2?

Unit 1: Polynomial arithmetic

Unit 2: complex numbers, unit 3: polynomial factorization, unit 4: polynomial division, unit 5: polynomial graphs, unit 6: rational exponents and radicals, unit 7: exponential models, unit 8: logarithms, unit 9: transformations of functions, unit 10: equations, unit 11: trigonometry, unit 12: modeling.

• Get started
• Pre-Algebra

A quicker path to better grades

We have gathered all your curriculum-based courses, assignments, hints, tests, and solutions in one easy-to-use place

• Integrated I
• Integrated II
• Integrated III

Can't find your textbook?

More math. less studying.

A personal private tutor for each student. Free from preassure and study anxiety.

• Find your textbook
• Math Solver

• Big Ideas Math Algebra 2, Virginia Edition, 2019

• Big Ideas Math Algebra 2, 2014

• Big Ideas Math Algebra 2 Texas

• Big Ideas Math Algebra 2 A Bridge to Success

• Core Connections Algebra 2, 2013

• Houghton Mifflin Harcourt Algebra 2, 2015

• McGraw Hill Glencoe Algebra 2, 2012

• McGraw Hill Glencoe Algebra 2 Texas, 2016

• McGraw Hill Glencoe Algebra 2, 2017

• Pearson Algebra 2 Common Core, 2011

• Pearson Algebra 2 Common Core, 2013

StudyMonkey

Your personal ai tutor.

Learn Smarter, Not Harder with AI

Introducing StudyMonkey, your AI-powered tutor .

StudyMonkey AI can tutor complex homework questions, enhance your essay writing and assess your work—all in seconds.

No more long all-nighters

24/7 solutions to questions you're stumped on and essays you procrastinated on.

No more stress and anxiety

Get all your assignments done with helpful answers in 10 seconds or less.

No more asking friends for help

StudyMonkey is your new smart bestie that will never ghost you.

No more staying after school

AI tutoring is available 24/7, on-demand when you need it most.

AI Tutor for any subject

American college testing (act), anthropology, advanced placement exams (ap exams), arabic language, archaeology, biochemistry, chartered financial analyst (cfa) exam, communications, computer science, certified public accountant (cpa) exam, cultural studies, cyber security, dental admission test (dat), discrete mathematics, earth science, elementary school, entrepreneurship, environmental science, farsi (persian) language, fundamentals of engineering (fe) exam, gender studies, graduate management admission test (gmat), graduate record examination (gre), greek language, hebrew language, high school entrance exam, high school, human geography, human resources, international english language testing system (ielts), information technology, international relations, independent school entrance exam (isee), linear algebra, linguistics, law school admission test (lsat), machine learning, master's degree, medical college admission test (mcat), meteorology, microbiology, middle school, national council licensure examination (nclex), national merit scholarship qualifying test (nmsqt), number theory, organic chemistry, project management professional (pmp), political science, portuguese language, probability, project management, preliminary sat (psat), public policy, public relations, russian language, scholastic assessment test (sat), social sciences, secondary school admission test (ssat), sustainability, swahili language, test of english as a foreign language (toefl), trigonometry, turkish language, united states medical licensing examination (usmle), web development, step-by-step guidance 24/7.

Receive step-by-step guidance & homework help for any homework problem & any subject 24/7

Ask any question

StudyMonkey supports every subject and every level of education from 1st grade to masters level.

Get an answer

StudyMonkey will give you an answer in seconds—multiple choice questions, short answers, and even an essays are supported!

Review your history

See your past questions and answers so you can review for tests and improve your grades.

It's not cheating...

You're just learning smarter than everyone else

How Can StudyMonkey Help You?

Hear from our happy students.

"The AI tutor is available 24/7, making it a convenient and accessible resource for students who need help with their homework at any time."

"Overall, StudyMonkey is an excellent tool for students looking to improve their understanding of homework topics and boost their academic success."

Upgrade to StudyMonkey Premium!

Why not upgrade to StudyMonkey Premium and get access to all features?

Algebra Examples

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&

Finding Intersections

11.1: Math Talk: When $f$ Meets $g$ (5 minutes)

CCSS Standards

Building On

• HSA-APR.B.3
• HSA-REI.C.7

Routines and Materials

Instructional Routines

• MLR8: Discussion Supports

In this activity, students have an opportunity to notice and make use of structure (MP7) in order to identify a point where the graphs of two given functions intersect. The work here leads directly into the next activity in which students use algebraic methods to identify all points of intersection between two polynomials.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Mentally identify a point where the graphs of the two functions intersect, if one exists.

\(f(x)=x\) and \(g(x)=3\)

\(j(x)=(x+3)(x-3)\) and \(k(x)=0\)

\(m(x)=(x+3)(x-3)\) and \(n(x)=(x-3)\)

\(p(x)=(x+5)(x-5)\) and \(q(x)=(x+3)(x-3)\)

Student Response

For access, consult one of our IM Certified Partners .

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\) ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\) ’s strategy?”
• “Do you agree or disagree? Why?”

If not brought up during the discussion, remind students that these are called systems of equations. Ask, “What are some ways you have solved systems of equations in the past?” (Graphing, substitution, elimination.) Tell students that in the next activity, they are going to think about how to use substitution to solve systems of equations in which at least one of the equations is a quadratic.

11.2: More Points of Intersection (20 minutes)

• HSA-REI.B.4
• Think Pair Share

The purpose of this lesson is for students to solve systems of equations involving quadratics. Unlike systems of linear equations, systems involving quadratic functions can have 0, 1, or 2 distinct solutions, in addition to the infinitely many solutions case. But as with linear equations, solutions can be identified from graphs as the \(x\) -values of the points where the lines cross. Students will think about solutions both algebraically and graphically in this lesson. Students begin the lesson by focusing on structure to identify possible solutions to a series of systems of equations (MP7). Building on this, students then combine their skills solving systems of equations using substitution and solving quadratic equations to identify all solutions to systems of equations involving at least one quadratic function. They will continue to use these skills in future lessons involving rational functions and polynomial identities.   The last activity is meant to make clear that while algebraic solving methods are useful, technology helps us identify solutions to systems of equations where the steps needed to solve by hand are less obvious. This activity also asks students to consider what a possible factor of a polynomial expression written in standard form could be, which will be the focus of the following lessons leading into the Remainder Theorem.

Arrange students in groups of 2. Display the equations \(a(x)=(x+2)(x-2)\) and \(b(x)=(x-2)\) for all to see, and ask students to try and solve the system without graphing. After quiet work time, have students share their work with their partner and reach agreement on the solutions. Invite 2–3 students to share their solution process before starting the activity.

After students have had 5 minutes to start working on the activity, ask them to pause, and display the following solution strategy for all to see, if it was not discussed earlier.

\(\displaystyle \begin {align*} (x+2)(x-2) &= (x-2) \\ (x+2)(x-2)-(x-2) &= 0 \\ (x-2)(x+2-1) &= 0 \\ (x-2)(x+1) &= 0 \\ \end {align*}\)

Ask, “What is happening at each step?” (Rewriting the equation so it is an expression equal to zero, then factoring the \((x-2)\) from each term.) Encourage students to look for structure as they are solving the remaining systems.

Since this activity was designed to be completed without technology, ask students to put away any devices.

For each pair of polynomials given, find all points of intersection of their graphs.

• \(c(x)=x^2-7\) and \(d(x)=2\)
• \(f(x)=(x+7)(x-4)\) and \(g(x)=x-4\)
• \(m(x)=(x+7)(x-4)\) and \(n(x)=(2x+5)(x-4)\)
• \(p(x)=(x+1)(x-8)\) and \(q(x)=(x+2)(x-4)\)

Are you ready for more?

Find all points of intersection of the graphs of the equations \(p(x)=(2x+3)(x-5)\) and \(q(x)=(x+5)(x+1)(x-3)\) . Use graphing technology to check your solutions.

Anticipated Misconceptions

Some students may divide by the same factor on each side of the equation during their solving process. For example, going from \((x+7)(x-4) = x-4\) in one step to \(x+7=1\) in the next. While this does result in one of the solutions, in this example,  \(x=\text-6\) , the other solution, \(x=4\) , is lost. Encourage these students to try solving without dividing by terms with an \(x\) in them to see both solutions. It may be helpful to ask, “What would happen if \(x\) was 4? Would it then make sense to divide by \(x-4\) ?” (No, because then we would be dividing each side by 0.)

The purpose of this discussion is for students to understand the different ways their classmates solved the questions in the activity. Select 1 previously identified student per question to share their solutions. Encourage students to ask clarifying questions about why different solving steps were used. After each student shares, ask if any students solved the question a different way, and invite those students to share their steps.

Conclude the discussion by asking students if it is possible for a system of equations with two quadratic functions to have 3 solutions. If possible, display a graph with two quadratics whose shapes can be manipulated to help convince students that 2 is the maximum number of solutions for distinct quadratic functions.

11.3: Graphing to Find Points of Intersection (10 minutes)

• HSA-REI.D.11

Building Towards

• MLR5: Co-Craft Questions

Required Materials

• Graphing technology

This activity revisits a polynomial students first encountered in an earlier lesson in this unit, relating the polynomials to the integers. The purpose of this activity is to make clear that technology allows us to identify solutions to equations that can be tedious to do by hand. The last question reinforces this point by asking students to write their own polynomial and find the \(x\) value that corresponds to a certain output, since they will need to estimate the solution from a graph.

The question about possible factors is meant to preview the work of the next several lessons in which students will identify factors of polynomial expressions written in standard form and eventually show that for a polynomial \(p(x)\) , if \(p(a)=0\) , then \((x-a)\) is a factor of \(p(x)\) .

Display functions \(p\) and \(q\) for all to see. Ask, “Without using a graphing calculator, what is the most points of intersection the two functions could have? The least?” Encourage students to make sketches to justify that 3 is the most and 1 is the least.

Provide access to devices that can run Desmos or other graphing technology.

Consider the functions \(p(x) = 5x^3+6x^2+4x\) and \(q(x) = 5640\) .

• Use graphing technology to find a value of \(x\) that makes \(p(x)=q(x)\) true.
• For the \(x\) -value at the point of intersection, what can you say about the value of \(5x^3+6x^2+4x-5640\) ?
• What does your answer suggest is a possible factor of \(5x^3+6x^2+4x-5640\) ?
• Write your own polynomial \(m(x)\)  of degree 3 or higher.
• Use graphing technology to estimate the values of \(x\) that make  \(m(x)= q(x)\) true.

If students have trouble using the value of \(5x^3+6x^2+4x-5640\) at the \(x\) -value where the functions intersect to find a possible factor of \(5x^3+6x^2+4x-5640\) , encourage them to think about the relationship between factors of an expression and \(x\) -values for which the value of the expression is 0.

The goal of this discussion is for students to see how to use technology to answer an algebraic question involving a higher degree polynomial. Invite students to share the polynomial \(m(x)\) that they wrote. If possible, display several versions of \(m(x)\) for all to see, and use the graph to estimate the \(x\) -values where \(m(x)=q(x)\) . If this is not possible, ask students to share the solutions they estimated. Students should understand that not all polynomials can be factored by hand, and that technology is a valuable tool for helping us interpret such polynomials.

Then ask students to suggest possible factors of \(5x^3+6x^2+4x-5640\) . After a few suggestions, ask students how they could prove something is a factor. It is important that students understand that for now, the only way to prove something like \((x-10)\) is a factor of \(5x^3+6x^2+4x-5640\) is to find the quadratic function that satisfies  \(5x^3+6x^2+4x-5640=(x-10)(ax^2+bx+c)\) , where \(a\) , \(b\) , and \(c\) are real numbers. Tell students that the next lessons are about how to do this.

Lesson Synthesis

The purpose of this synthesis is for students to reflect on the similarities and differences when solving systems of equations in which both equations are linear and when solving systems of equations in which one equation is quadratic and the other is linear.

Display the following systems for all to see:

\(\displaystyle \begin {align*} y &=x+5 \\ y &= 5-x \\ \end {align*}\)

\(\displaystyle \begin {align*} y &=x+5 \\ y &= (x-2)(x+5) \\ \end {align*}\)

Ask students to solve both systems and then write a brief summary of how solving systems with quadratic equations is similar and different from solving systems with only linear equations. If time allows, invite students to share some of the things they wrote. Highlight any students that note the difference in the number of possible solutions and those who notice that the steps used to solve for \(x\) can be very similar.

11.4: Cool-down - Find Some More Points (5 minutes)

Student lesson summary.

When asked to find all values of \(x\) that make an equation like \((x+4)(x-8)=(2-x)(x-8)\) true, one way to consider the question is to ask where the graphs of the functions \(f(x)=(x+4)(x-8)\) and \(g(x)=(2-x)(x-8)\) intersect.

Description: <p>Functions f of x and g of x on an x y coordinate plane, no grid. Horizontal axis from negative 6 to 9, by 1's. Vertical axis from negative 50 to 10, by 10s. Starting in quadrant 2, f of x moves downwards, passing through negative 4 comma 0, intersects g of x at negative 1 comma negative 27, curves and moves upwards, intersecting g of x, and passing through 8 comma 0. Starting in quadrant 3, g of x moves upwards and to the right, intersects f of x at negative 1 comma negative 27, curves and moves downwards, intersecting f of x and 8 comma 0. <br>  </p>

Since the coordinate of any point of intersection has the form \((a,f(a))=(a,g(a))\) , these points must make \(f(x)=g(x)\) true when \(x=a\) . In our example, we can tell from the graph that both \(x=\text-1\) and \(x=8\) are solutions to the original equation.

We can also use algebra to identify solutions to \((x+4)(x-8)=(2-x)(x-8)\) by rearranging and then recognizing that both parts have a factor of \((x-8)\) in common:

\(\displaystyle \begin{align*} (x+4)(x-8)&=(2-x)(x-8)\\ (x+4)(x-8)-(2-x)(x-8)&=0\\ (x-8)(x+4-2+x)&=0\\ (x-8)(2x+2)&=0\\ x&=8, \text-1\\ \end{align*}\)

For polynomials created to model specific situations that have a more messy structure, solving without using technology can be challenging, especially because the graphs of two polynomials can intersect at multiple points because of the way they curve. Fortunately, this type of solving challenge is one that computer algebra systems are usually very good at, leaving the interpretation of the solution up to humans.

View Answer Keys

View the correct answers for activities in the learning path.

This procedure is for activities that are not provided by an app in the toolbar.

Some MindTap courses contain only activities provided by apps.

• Click an activity in the learning path.

Jordan stations 2 firefighting helicopters in Cyprus to help as summer fire season arrives

Jordan has stationed two helicopters in Cyprus to help the Mediterranean island nation combat potential forest fires until it can lease firefighting aircraft of its own for the summer fire season

NICOSIA, Cyprus — Jordan on Monday stationed two helicopters in Cyprus to help the Mediterranean island nation combat potential forest fires until it can lease firefighting aircraft of its own for the summer fire season.

Cypriot Environment Minister Maria Panayiotou said the Cypriot government considers firefighting and prevention a national security matter and is also moving to purchase 10 aircraft for that reason.

Panayiotou welcomed the 18 Royal Jordanian Air Force technical staff, crew and pilots who will “stand shoulder to shoulder with us in aerial firefighting.” The Cypriot government will cover the Jordanian mission’s expenses through its deployment. It was unclear how long that will be.

The minister said the Jordanian crews will undergo immediate training to adapt to local conditions and will become operationally ready in the next few days.

The Jordanian helicopters will bolster a fleet of at least three other aircraft from Cyprus’ police and National Guard, including a helicopter from two military bases that Britain maintains on the island, that will be on round-the-clock call.

The Jordanian deployment comes after a Cypriot proposal to set up a regional fire fighting hub in Cyprus from which aircraft and other technology could be dispatched to help put out fires in neighboring countries . Two years ago, Jordan’s King Abdullah II replied by offering to station firefighting aircraft in Cyprus through the summer fire season.

Cyprus, Greece, Jordan and Israel have often assisted each other in recent years by sending firefighting teams, gear and aircraft to help combat massive wildfires.