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Making Sense of Extraneous Solutions
Article in Mathematics Teacher · February 2013
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Do you always have to check your answers
lways check your answers when
solving any radical equation.” While
observing a class as a mentor, I overheard this response from a student
teacher to a student’s question, “Do
we need to check our answers to this [cube root]
equation?” Good teachers will always emphasize
checking work after solving a problem, but this particular question referred to the possible introduction of an extraneous root. The solution of a radical
equation involving only cube roots, where checking
for extraneous algebraic solutions is not required,
was not discussed by the student teacher; there was
no discussion or examination of function domain.
The underlying reasoning and sense making of
extraneous solutions did not occur at any point in
the lesson. As a result, I began to develop a technology-based lesson using the TI-Nspire™ CAS.
Principles and Standards for School Mathematics
(NCTM 2000) states, “Technology is essential in
teaching and learning mathematics; it influences the
mathematics that is taught and enhances students’
learning” (p. 11). The lesson in which I observed
the teacher-student exchange quoted above took
place in a second-year algebra classroom. It focused
solely on procedures and used technology simply as
a computational tool for checking answers to equations containing square and cube roots. The focus
on reasoning and sense making with technology in
the lesson presented here will enable students to do
more than just carry out procedures; they will be
able to understand the procedures and know how
452 MatheMatics teacher | Vol. 106, No. 6 • February 2013
Copyright © 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
when solving a radical equation?
Jeremy s. Zelkowski
the procedures might be used in different situations
while interpreting the results (NCTM 2009).
The lesson includes three key elements of reasoning and sense making with functions: (1) using
multiple representations of functions as a means of
demonstrating mathematical flexibility in problem
solving; (2) modeling by using families of functions;
and (3) analyzing the effects of parameters (NCTM
2010). According to the Programme for International
Student Assessment (PISA 2007), American students
lag behind international students in their ability to
analyze, reason, communicate, solve, and interpret a
variety of mathematical problems. The following lesson
allows teachers to use a pedagogical approach to teaching mathematics through problem solving (see Schoen
2003) and also address these PISA areas of concern.
By exploring the following two problems, students will improve their ability to understand and
solve problems involving radicals:
Problem 1: Determine an equation with a single
radical that, when solved algebraically, will
yield one unique real solution and two extraneous solutions.
Problem 2: Without working out this problem
with paper and pencil, predict how many real
and extraneous solutions would be obtained
when solving 5x + 3 – 2 = 2x + 3 algebraically.
For each problem, explain your reasoning in
support of your responses.
Vol. 106, No. 6 • February 2013 | MatheMatics teacher 453
Fig. 2 Graphing the negation of the radical function
reveals the extraneous algebraic solution.
Fig. 1 the power of the graphical solutions (a and c) and
the cas solution (b) to equations 1 and 2 adds to student
understanding and establishes new connections.
When students demonstrate the algebraic solutions on the TI-Nspire CAS, the handheld tool gives
this warning: Operation might introduce false solutions, a signal for students to explore further.
In figure 1c, the graph indicates that equation
2 has only one real solution, x = 2; thus, the other
algebraic solution (x = –1) must be extraneous.
Students should explore how the graph can be used
to explain the extraneous solution, which is visible
when –f1(x) is graphed and an intersection point
occurs at (–1, –1) (see fig. 2). Hence, the x = –1
extraneous solution now appears graphically.
Students must learn that squaring both sides
of the equation, even one as simple as x = 2, will
introduce extraneous roots. They need to realize
that squaring to undo the square-root function
means that both cases of the radical function need
to be considered. For example, in the equation
±5x + 2 = x, both
( x + 2)
= x 2 and − ( x + 2)
result in x + 2 = x2 → x2 – x – 2 = 0 .
SOLVING PROBLEM 1
At the start of the lesson, the teacher and students
can work through skill-focused problems similar to
equations 1 and 2, as needed.
Equation 1: 5x – 2 = 5
Equation 2: 5x + 2 = x
Then students can begin to make connections
between solution methods by using paper and
pencil or a computer algebra system or by creating
a graph. Alternatively, the teacher may elect to
begin the TI-Nspire lesson through discovery. The
CAS eliminates computational errors and allows
students to focus on content and connections
between multiple representations without losing
the opportunity to demonstrate problem-solving
ability and algebra skills (see fig. 1 for graphical
and CAS solutions).
454 MatheMatics teacher | Vol. 106, No. 6 • February 2013
EXAMINING FAMILIES OF EQUATIONS
The interactive geometry capability of the TINspire (with or without CAS) can take the lesson
to a higher cognitive level. As figure 3 shows,
students are able to manipulate the linear function f1(x) dynamically. They can manipulate the
y-intercept and slope independently. To manipulate the y-intercept, students move the cursor
arrow near the y-intercept (see the crosshairs near
the origin in fig. 3a) and grab the line (center of
touchpad). To change the slope, students move the
cursor arrow near the ends of the line (see fig. 3b,
lower right) and grab.
The parameters a, b, and c of the radical function can be changed incrementally. Figure 3 shows
a student’s work in shifting the radical function,
f2(x), to the right three units, thus eliminating the
Fig. 4 This student initially attempted to solve problem 1
by graphing a radical and a quadratic function. The
parabola is tangent to the radical function and intersects
the negation of the radical function twice.
Fig. 3 A student changes parameters to transform both
the radical and the linear function. Going from (a) to (b),
the slope of the line and the x-intercept of the radical
function have been adjusted.
intersection with the linear function. The student
also changed the slope of f1(x) from 1 to –0.5.
These transformations produce two functions that,
when set equal to each other, will produce an equation with two extraneous algebraic solutions. The
dynamic capability empowers students to build
their own mathematical understanding as well as to
exercise their reasoning and sense making.
However, the possibility of complex solutions of
equations from the family of functions generated
by f1(x) and f2(x) in figure 3 can easily be overlooked by students. Teachers need to be ready to
incorporate guiding questions such as these: What
additional possibilities exist for solution sets that
we have not yet examined graphically? What are
the graphical representations for each combination
of solutions? Are there possibilities that we may be
overlooking? What would happen if f1(x), f2(x),
and –f2(x) do not intersect at all?
After discussing the complex solution possibilities, students should begin to think about the
necessary conditions that will add a third possible
solution, as required by problem 1 of the lesson.
The solution sets that can be generated by the family of functions explored in figure 3 have only two
algebraic solutions (real, extraneous, or complex).
This will help differentiate the differences between
Fig. 5 A student’s algebraic solution to the proposed
graphical solution seen in figure 4 uses CAS.
extraneous and complex solutions—which students
sometimes think are the same thing.
When I have used this lesson, most students
gravitate toward sketching a graphical solution for
problem 1. At times, students use only polynomial
functions and try a linear function composed with
the basic radical function. This approach limits the
Without an equation, students have trouble
making a start on this problem. Trial and error or
guess and check usually do not yield a solution.
One student proposed a possible graphical solution
(see fig. 4), an equation that would involve the
radical function previously explored and a quadratic function. This choice shows great graphical
intuition, but this solution needs further examination. The intersection near or on the y-axis must be
at a point of tangency. This proposed solution leads
to one piece of mathematical understanding needed
for a solution.
Vol. 106, No. 6 • February 2013 | Mathematics Teacher 455
Fig. 6 Students’ only recourse to solving an equation such
as 2.3 sin x = 5x — 1 is to use a graph or a CAS.
Students can then explore this proposed solution
by solving an equation (see the CAS solution shown
in fig. 5). By using the CAS to solve this proposed
equation algebraically, students realize that four
answers result: x ≈ –1.86, –0.25, 0, and 2.11. The
point (in fig. 4) near (0, 2) is worthy of discussion
because it ended up not being a point of tangency.
The two functions intersect twice (at x = 0 and at
x ≈ –0.25) in the region near the positive y-axis.
Students need to make two important discoveries in the lesson to be able to solve problem 1. First,
they must realize that an equation with a single
cube-root radical will not generate any extraneous
solutions; only an even-root radical will. (A fuller
discussion of this concept is available at the website
provided at the end of the article.) Second, students
must discover that after they square both sides of
the equation, the result must generate three solutions. In other words, squaring the desired equation
must produce a cubic polynomial.
Once students make the mathematical connections needed, they can use the TI-Nspire capabilities (CAS, parameterization, transformation of
linear functions) to generate a correct solution to
problem 1. For example, 6x3 – 4x = x – 0.5 yields
one solution, but there are an infinite number of
answers involving any radical with an even index.
At this point in the lesson, teachers could ask questions such as these:
• What would happen if we try to generate a solution that includes a trigonometric or a transcendental function?
• What would happen if we try to generate an
equation with only integer or rational solutions?
One student used the TI-Nspire to generate a
fairly convincing graphical argument to a trigonometric-transcendental question (see fig. 6).
Some students have tried to generate the point of
tangency seen in figure 4 or integer-only solutions
by reversing the problem—that is, by starting with
456 Mathematics Teacher | Vol. 106, No. 6 • February 2013
Fig. 7 A student uses reversibility to generate a solution to
problem 1 (a) and then verifies the solution graphically (b).
a factored form of the equation and working backward algebraically. For example, one student started
with x • x • (x + 2) • (x – 3) = 0, which has a double
root at x = 0 and could be a solution to the graphical
representation seen in figure 4. The student first
removed parentheses and then ended up with the
equation 7x3 + 6x2 = –x2 (see fig. 7). An examination of the graphical representation made clear that
the x2 should be on the right side of the equation and
should be negative. Although this double root is not
a point of tangency as defined in geometry or calculus, the student found a correct solution to problem 1
with integer-only algebraic solutions.
This lesson thus helps develop students’ ability
to use mathematical flexibility and reversibility in
problem solving (Krutetskii 1976; see also Rachlin
1985). Making connections between algebraic and
graphical solutions also demonstrates the use of
mathematical flexibility in problem solving. The
family of equations of the form a5x + b + c =
mx + B cannot be used to solve problem 1. Deep
understanding of the mathematical connections
between algebraic and graphical solutions from
this family of functions presents an environment
conducive to reasoning and sense making. Both are
needed to move to mathematical problems requiring higher levels of cognitive effort.
Fig. 9 The graphical representation presents four
solutions to problem 2, three of which are extraneous.
Fig. 8 Students can solve problem 2 using a graph (a);
the other branch of the sideways parabola shows an
extraneous root (b).
SOLVING PROBLEM 2
The beauty of this problem is that it allows for
different approaches and different solutions. The
previous class discussions, leading questions, and
equations explore reflecting the simple square-root
function over the x-axis, thus creating a parabola
opening to the right or left with its vertex on the
x-axis. The lesson does not guide students through
the examination of a square-root function that
crosses the x-axis, as is the case with problem 2,
or one that lies completely above the x-axis. If we
plot the left side of the equation in problem 2 or
visualize y = 1x shifted to the left three units and
down two units, we see that this radical function
crosses the x-axis (see fig. 8). Student responses
to this problem have generally fit into one of two
Most students’ first response includes shifting
the radical and the linear functions up two units.
Essentially, this is what we would expect or want
students to do first algebraically—add 2 to both
sides of the equation first, then square, and so on.
Thus, students accurately predict one real solution
and one extraneous solution.
The second type of response generally comes
from creating the parabola seen in figure 8b. I
want students to examine and understand—
algebraically and graphically—what will happen if
we square both sides of the equation first. Doing so
is a common first step by some students when they
see a square root sign. Mathematically, squaring
first is not as efficient. Only a few students have
ever submitted an answer such as that shown in
figure 9. This solution accurately depicts squaring both sides first and eventually solving a quartic
polynomial equation that will generate a real solution and three extraneous solutions.
Students must consider the four cases of the
radical equation ± [±5x + 3 – 2] = 2x + 3: (1) 5x + 3
– 2 = 2x + 3; (2) –5x + 3 – 2 = 2x + 3; (3) –[5x + 3]
– 2] = 2x + 3; and (4) –[–5x + 3 – 2] = 2x + 3. Cases
1 and 3 yield the equation –45x + 3 = 4x2 + 11x + 2,
and cases 2 and 4 yield the equation 45x + 3 = 4x2
+ 11x + 2. We now solve ±45x + 3 = 4x2 + 11x + 2,
which both produce the quartic polynomial equation 16(x + 3) = 16x4 + 88x3 + 137x2 + 44x + 4, with
solutions at x = –2.75, –2, (–341 – 3)/8 ≈ –1.1754,
and (341 – 3)/8 ≈ 0.4254.
In figure 9, the highlighted point depicts the
only real solution at x = −2 and represents case 1,
f1(x) = f2(x). The three extraneous solutions are
depicted with the dotted functions intersecting the
linear function. Case 2, f3(x) = f2(x), yields x =
−2.75. Case 3, f5(x) = f2(x), yields x ≈ −1.1754.
Case 4, f4(x) = f2(x), yields x ≈ 0.4254.
By not providing a direct path or direction to
solve problem 2, I am able to assess students on a
higher cognitive level (Stein et al. 2000).Thus, I do
not provide an in-depth exploration of this problem
type during the actual lesson. This approach also
serves as a way to create discussion about the
mathematics of this lesson after the assignment
has been collected.
This lesson was originally planned for secondary
school mathematics preservice teachers and their
Vol. 106, No. 6 • February 2013 | Mathematics Teacher 457
development of Technological Pedagogical and Content Knowledge (TPACK; see Mishra and Koehler
2006), mathematical reversibility, flexibility, and
reasoning and sense making.
It has generally been acknowledged that it
takes three to five years of training in technology
for teachers to be comfortable and knowledgeable
enough to use it regularly in teaching (Dwyer,
Ringstaff, and Sandholtz 1991; Means and Olson
1994). That was true for a different generation of
teachers. In today’s technologically driven world,
three to five years is too long. In theory, because
today’s generation of secondary school mathematics teachers grew up with technology in their hands
often and early, this time should be cut in half (see
Leatham 2007; Norton, McRobbie, and Cooper
2000; Zelkowski 2011).
This lesson is just one that can help in the development of TPACK as well as content knowledge.
Moreover, this approach to a topic traditionally
taught through rote procedure can be extremely
engaging in the high school classroom.
Dwyer, David C., Cathy Ringstaff, and Judy H. Sandholtz. 1991. “Changes in Teachers’ Beliefs and
Practices in Technology-Rich Classrooms.” Educational Leadership 48 (8): 45–52.
Krutetskii, Vadim A. 1976. The Psychology of Mathematical Abilities in School Children. Trans. from the Russian by Joan Tell; edited by Jeremy Kilpatrick and
Izaak Wirszup. Chicago: University of Chicago Press.
Leatham, Keith R. 2007. “Pre-service Secondary
Mathematics Teachers’ Beliefs about the Nature of
Technology in the Classroom.” Canadian Journal
of Science, Mathematics, and Technology Education 7
Means, Barbara, and Kerry Olson. 1994. “Tomorrow’s
Schools: Technology and Reform in Partnership.” In
Technology and Education Reform, edited by Barbara
Means, pp. 191–222. San Francisco: Jossey-Bass.
Mishra, Punya, and Matthew J. Koehler. 2006. “Technological, Pedagogical, and Content Knowledge: A
Framework for Integrating Technology in Teacher
Knowledge.” Teachers College Record 108 (6):
National Council of Teachers of Mathematics
(NCTM). 2000. Principles and Standards for School
Mathematics. Reston, VA: NCTM.
———. 2009. Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: NCTM.
———. 2010. Focus in High School Mathematics: Reasoning and Sense Making in Algebra. Reston, VA: NCTM.
Norton, Stephen, Campbell J. McRobbie, and Tom J.
Cooper. 2000. “Exploring Secondary Mathematics Teachers’ Reasons for Not Using Computers
in Their Teaching: Five Case Studies.” Journal of
458 Mathematics Teacher | Vol. 106, No. 6 • February 2013
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Research on Computing in Education 33 (1): 87–109.
Programme for International Student Assessment
(PISA). 2007. PISA 2006: Science Competencies
for Tomorrow’s World. Paris: Organisation for
Economic Co-operation and Development. http://
Rachlin, Sidney L. 1985. “The Development of Problem-Solving Processes in a Heterogeneous EighthGrade Algebra Class.” Paper presented at the annual
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