9.2.4C Equations Containing Radical Expressions
Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.
For example: The equation $\sqrt{x9}=9\sqrt{x}$ may be solved by squaring both sides to obtain x  9 = 81x, which has the solution $x=\frac{9}{80}$. However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.
Another example: Solve $\sqrt[3]{x+1}=5$.
Overview
Standard 9.2.4 Essential Understandings
In this standard, students learn to model realworld mathematical situations using linear, quadratic, exponential and n^{th} root functions. In order to do so, they need to understand the characteristics of each type of function: linear functions have a constant rate of change; quadratic functions have a linear rate of change (the rate of change of the rate of change, or the 2^{nd} difference, is constant); and exponential functions have a rate of change that is proportional to the value of the function (the output is multiplied by a constant factor).
Students need to be able to choose from a variety of methods to solve these functions, whether symbolically or graphically. Once solutions are found, students need to be able to determine the reasonableness of an answer given the realworld context for the function  a particular solution may not be applicable in the original context.
Students need to understand the real number system, including the subsets of natural and whole numbers, integers, rational and irrational numbers, and that many of these numbers were invented to solve equations. They need to know how the real number system fits into the nonreal complex number system, and how operations in the nonreal complex number system correspond to those of the real number system. Students need to understand how a solution in the nonreal complex number system is graphed, and what that solution means in terms of a given context for a quadratic equation.
Students need to be comfortable in using a graphing calculator or other graphing utilities to find solutions and describe characteristics of linear, quadratic and exponential functions. They should know how to use the table and graphing features to check solutions for these functions.
All Standard Benchmarks
9.2.4.1
Represent relationships in various contexts using quadratic equations and inequalities. Solve quadratic equations and inequalities by appropriate methods including factoring, completing the square, graphing and the quadratic formula. Find nonreal complex roots when they exist. Recognize that a particular solution may not be applicable in the original context. Know how to use calculators, graphing utilities or other technology to solve quadratic equations and inequalities.
Example: A diver jumps from a 20 meter platform with an upward velocity of 3 meters per second. In finding the time at which the diver hits the surface of the water, the resulting quadratic equation has a positive and a negative solution. The negative solution should be discarded because of the context.
9.2.4.2
Represent relationships in various contexts using equations involving exponential functions; solve these equations graphically or numerically. Know how to use calculators, graphing utilities or other technology to solve these equations.
9.2.4.3
Recognize that to solve certain equations, number systems need to be extended from whole numbers to integers, from integers to rational numbers, from rational numbers to real numbers, and from real numbers to complex numbers. In particular, nonreal complex numbers are needed to solve some quadratic equations with real coefficients.
9.2.4.4
Represent relationships in various contexts using systems of linear inequalities; solve them graphically. Indicate which parts of the boundary are included in and excluded from the solution set using solid and dotted lines.
9.2.4.5
Solve linear programming problems in two variables using graphical methods.
9.2.4.6
Represent relationships in various contexts using absolute value inequalities in two variables; solve them graphically.
Example: If a pipe is to be cut to a length of 5 meters, accurate to within a tenth of its diameter, the relationship between the length x of the pipe and its diameter y satisfies the inequality  x  5 ≤ 0.1y.
9.2.4.7
Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.
Example: The equation $\sqrt{x9}=9\sqrt{x}$ may be solved by squaring both sides to obtain x  9 = 81x, which has the solution $x=\frac{9}{80}$. However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.
Another example: Solve $\sqrt[3]{x+1}=5$.
9.2.4.8 Assess the reasonableness of a solution in its given context and compare the solution to appropriate graphical or numerical estimates; interpret a solution in the original context.
9.2.4 Algebra Benchmark Group C  Equations containing Radical Expressions
9.2.4.7
Solve equations that contain radical expressions. Recognize that extraneous solutions may arise when using symbolic methods.
For example: The equation $\sqrt{x9}=9\sqrt{x}$ may be solved by squaring both sides to obtain x  9 = 81x, which has the solution $x=\frac{9}{80}$. However, this is not a solution of the original equation, so it is an extraneous solution that should be discarded. The original equation has no solution in this case.
Another example: Solve $\sqrt[3]{x+1}=5$.
What students should know and be able to do [at a mastery level] related to these benchmarks:
Use a calculator to find decimal approximations for radical expressions such as $\sqrt{7},4\sqrt{3},\sqrt[3]{17},\sqrt[5]{7}$ and $7+2\sqrt{5}$;
Solve equations involving radicals by taking both sides of an equation to an exponent. Furthermore, students should know that this procedure does not always generate an equivalent equation.
Work from previous grades that supports this new learning includes:
 Know how to evaluate radical expressions with numerals under the radicand.
NCTM Standards
Understand patterns, relations, and functions:
 Generalize patterns using explicitly defined and recursively defined functions;
 Understand relations and functions and select, convert flexibly among, and use various representations for them;
 Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior;
 Understand and perform transformations, such as arithmetically combining, composing, and inverting commonly used functions, using technology to perform such operations on morecomplicated symbolic expressions;
 Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions;
 Interpret representations of functions of two variables.
Represent and analyze mathematical situations and structures using algebraic symbols:
 Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations;
 Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency  mentally or with paper and pencil in simple cases and using technology in all cases;
 Use symbolic algebra to represent and explain mathematical relationships;
 Use a variety of symbolic representations, including recursive and parametric equations, for functions and relations;
 Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.
Use mathematical models to represent and understand quantitative relationships:
 Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships;
 Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts;
 Draw reasonable conclusions about a situation being modeled.
Analyze change in various context:
 Approximate and interpret rates of change from graphical and numerical data.
Common Core State Standards (CCSM)
ASSE (Seeing Structure In Expressions) Write expressions in equivalent forms to solve problems.
ASSE.3.Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
ASSE.3.a. Factor a quadratic expression to reveal the zeros of the function it defines.
ASSE.3.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
ASSE.3.c. Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^{t} can be rewritten as (1.15^{1/12})^{12t} ≈ 1.012^{12t} to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
 ASSE.4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
ACED (Creating Equations) Create equations that describe numbers or relationships.
 ACED.1.Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
 ACED.2.Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
 ACED.3.Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
AREI (Reasoning with Equations and Inequalities) Understand solving equations as a process of reasoning and explain the reasoning.
 AREI.1.Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
 AREI.2.Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
AREI (Reasoning with Equations and Inequalities) Solve equations and inequalities in one variable.
 AREI.3.Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
 AREI.4.Solve quadratic equations in one variable.
AREI.4.a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x  p)^{2} = q that has the same solutions. Derive the quadratic formula from this form.
AREI.4.b. Solve quadratic equations by inspection (e.g., for x^{2} = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them asa ± bi for real numbers a and b.
AREI (Reasoning with Equations and Inequalities) Represent and solve equations and inequalities graphically.
 AREI.10.Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
 AREI.11.Explain why thexcoordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
FLE (Linear, Quadratic, & Exponential Models) Construct and compare linear, quadratic, and exponential models and solve problems.
 FLE.4.For exponential models, express as a logarithm the solution to ab^{ct} = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
FLE (Linear, Quadratic, & Exponential Models) Interpret expressions for functions in terms of the situation they model.
 FLE.5.Interpret the parameters in a linear or exponential function in terms of a context.
Misconceptions
Student Misconceptions and Common Errors
Students do not know how to find a decimal approximation for radical expressions using their calculators.
 Students will square expressions involving radicals incorrectly. For example, $\left ( 3\sqrt{x} \right )^{2}=3x$ or $\left ( \sqrt{t}+5 \right )^{2}=t+25$.
Vignette
In the Classroom
 When students learn to take both sides of an equation to an exponent, they do not always end up with an equivalent equation. A teacher should show this to students as they solve equations involving radicals. The table showing the symbolic steps in solving the equation $3\sqrt{x2}+5=2x1$ is shown with a column showing a graphical representation of the equation.
 equation  reason for step  graph 
 $3\sqrt{x2}+5=2x1$  original equation  
step 1  $3\sqrt{x2}=2x6$  addition property of equality  
step 2  $9\left ( x2 \right )=4x^{2}24x+36$  square both sides  
step 3  $9x18=4x^{2}24x+36$  distributive property  
step 4  $0=4x^{2}33x+54$  addition property of equality  
step 5  $0=\left ( x6 \right )\left ( 4x9 \right )$  factoring  
step 6  $x=6$ or $x=\frac{9}{4}$  zeroproduct property 

The students should notice that the graphs of the original equation and the graph after step 1 resulted in an equivalent equation since the graph shows the same solution set. The graph after step 2 shows another solution at $x=\frac{9}{4}$. This shows that squaring both sides of an equation does not always generate an equivalent equation. Teachers need to use graphical representations to help students make sense of unusual situations. The symbolic representation can help to make sense why the extra solution occurred. If $x=\frac{9}{4}$ is substituted into the equation in step 1, the equation would be $3\sqrt{\frac{9}{4}2}=2\left ( \frac{9}{4} \right )6$, which simplifies to $\frac{3}{2}=\frac{3}{2}$. This equation is not true, but becomes true when both sides are squared.
Resources
Teacher Notes
 Teachers need to demonstrate how to use technology and how to find decimal approximations mentally and with their calculators. Many students will enter the expression $\sqrt{3}+6$ into their graphing calculator incorrectly as $\sqrt{\left ( 3+6 \right )}$ and get 3 as an output instead of 7.73. Teachers should get students to constantly check the results of calculations with mental estimations. Students who enter $\sqrt{3}+6$ should be able to reason that the result should be greater than 6. A better estimate for the expression would be slightly smaller than 8 since $\sqrt{3}$ is a little less than $\sqrt{4}$ which is equal to 2. Students typically do not estimate calculations unless the teacher makes this practice a classroom norm.
 When students make mistakes involving radicals such as $\left ( 3\sqrt{x} \right )^{2}=3x$, teachers need to emphasize that the expression $\left ( 3\sqrt{x} \right )^{2}$ means $3\sqrt{x}\cdot 3\sqrt{x}$. Students may also need to be reminded that $3\sqrt{x}$ means $3\cdot \sqrt{x}$ so they can use the commutative property of multiplication to get the equivalent expression $3\cdot3\cdot\sqrt{x}\cdot\sqrt{x}$, which can then be simplified to 9x.
 Students will not discard extraneous solutions to equations involving radical expressions. An example is shown below.
 equation  reason for step 
 $3\sqrt{x2}+5=2x1$  original equation 
step 1  $3\sqrt{x2}=2x6$  addition property of equality 
step 2  $9\left ( x2 \right )=4x^{2}24x+36$  square both sides 
step 3  $9x18=4x^{2}24x+36$  distributive property 
step 4  $0=4x^{2}33x+54$  addition property of equality 
step 5  $0=\left ( x6 \right )\left ( 4x9 \right )$  factoring 
step 6  $x=6$ or $x=\frac{9}{4}$  zeroproduct property 
Students who solve equations like this are doing every step correctly and the justification for each step is correct. Students struggle with thinking that they did every step of the equation solving process correctly but still did not get full credit for their response because they did not check if their solutions worked. $x=\frac{9}{4}$ is an extraneous solution. Teachers often give the advice to "always check your solution" when solving equations involving radicals, but this could be simply memorized by students if the reasons are not explained.
radical: The √ symbol, which is used to indicate square roots or nth roots.
n^{th} power: The n^{th} power for a given base is the number that holds true for any given value of n. For example, 2^{n} > 8 would hold true for any value of n > 3.
extraneous solution / spurious solution: A solution of a simplified version of an equation that does not satisfy the original equation. Watch out for extraneous solutions when solving equations with a variable in the denominator of a rational expression, with a variable in the argument of a logarithm, or a variable as the radicand in an nth root when n is an even number.
Example: \begin{align*}x6 & = \sqrt{x}\\x^{2}12x+36 & = x\\x^{2}13x+36 & = 0\\\left ( x9 \right )\left ( x4 \right ) & = 0\\x=9,x=4\end{align*}
. . . or so it seems. In fact, though, only $x=9$ works in the original equation. $x=4$ does not work (try it!). It is extraneous.
Reflection  Critical Questions regarding the teaching and learning of these benchmarks
 Do students understand the concept of equivalent equations?
 Are students flexible in their use of things that they can do to both sides of an equation that result in equivalent equations?
Materials
 Burke, M. J., Hodgson, T., Kehle, P., Mara, P., & Resek, D. (2006). Growing Balloons. In Burke, M. J., & House, P. A. (Eds.), Navigating through mathematical connections in grades 912 (pp. 1319, 8892). Reston, VA: National Council of Teachers of Mathematics.
This resource provides background information on teaching about a realworld scenario involving radical expressions, including sections on discussion, assessment and where to go next in instruction:
 Common Core State Standards
http://www.corestandards.org/thestandards/mathematics  Ruddell, M.R., & Shearer, B.A. (2002). "Extraordinary," "tremendous," "exhilarating, "magnificent": Middle school atrisk students become avid word learners with the vocabulary selfcollection strategy (VSS). Journal of Adolescent & Adult Literacy, 45, 352363.
Assessment
1.
A result of global warming is that the ice of some glaciers is melting. Twelve years after the ice disappears, tiny plants, called lichen, start to grow on the rocks.
Each lichen grows approximately in the shape of a circle.
The relationship between the diameter of this circle and the age of the lichen can be approximated with the formula: $d=7.0\times\sqrt{\left ( t12 \right )}\; \; for\; t\geq 12$, where $d$ represents the diameter of the lichen in millimetres, and $t$ represents the number of years after the ice has disappeared.
Question 1:
Using the formula, calculate the diameter of the lichen, 16 years after the ice disappeared.
Show your calculation.
Question 2:
Ann measured the diameter of some lichen and found it was 35 millimetres.
How many years ago did the ice disappear at this spot?
Show your caluclation.
DOK Level: 3 / Cognitive Level: Application
Answers:
Q1: 14 mm
Q2: 37 years
Source: Programme for International Student Assessment (PISA) released item  M047 http://www.oecd.org/dataoecd/14/10/38709418.pdf
2.
Solve the following equation: $\sqrt{y1}+7=y+4$
A. y=2
B. y=2 or y=5
C. y=2
D. y=9
E. y=5
DOK Level: 2 / Cognitive Level: Application
Answer: e
Differentiation
 The most important concept around this standard is that taking both sides of an equation to the n^{th} power results in an equation that may or may not be equivalent. The problems most students run into when solving equations involving radicals is the distributive property when they take a polynomial to a power and the introduction of extraneous solutions. Teachers can begin with problems that allow students to use area models to multiply polynomials and use graphs to demonstrate extraneous solutions.
 Giving students an opportunity to use language to describe mathematical objects is crucial for deeper understanding of these objects. The activity below is called "Which one does not belong?" The object is for students to look at the four objects and decide which one does not belong and explain in words why it does not.
Which one does not belong?
A
$\sqrt{11}$  B
$\sqrt[3]{8}$ 
C
$\sqrt{36}$  D
$\sqrt{100}$ 
There are many different correct answers based on what the students see. Some students might explain that B does not belong because it is a cube root instead of a square root. Another student might explain that B does not belong because it is the only negative number. Others might exclude A because it is the only irrational number. There are many different solutions and the descriptions highlight important ideas of the objects the teacher selects to put in the boxes.
An example of a game highlighting radical equations is provided below.
Which one does not belong?
A
$3=\sqrt{x+4}$  B
$\sqrt{y1}+7=y+4$ 
C
$3\sqrt[3]{x+3}+5=11$  D
$7\sqrt{x+4}=21$ 
 Ask students to solve the following problem. Find [mathx^{2}$ if $x$ satisfies the equation:
$\sqrt[3]{x+9}\sqrt[3]{x9}=3$
Parents/Admin
Administrative/Peer Classroom Observation:
Students are:  Teachers are: 
able to explain that taking both sides of an equation to the n^{th} power usually results in an equivalent equation.  discussing with students the functions that can be done to both sides of an equation that preserve equivalence as well as the side effects for each function that result in nonequivalent equations. 
checking to see if their equations have extraneous solutions.  using multiple representations to discuss how extraneous solutions occur. 
able to justify each step of a series of equivalent equations.  modeling notation to emphasize the concept of equivalent equations. 
Parent Resources
 Function Transformations
 Use an interactive graphing tool to match functions with transformations of these functions. Questions and activity sheets are provided.