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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to recognize square root equations and extraneous solutions, be able to sove square root equations.

Overview

  • Square Root Equations And Extraneous Solutions
  • Method For Solving Square Root Equations

Square root equations and extraneous solutions

Square root equation

A square root equation is an equation that contains a variable under a square root sign. The fact that x · x = ( x ) 2 = x suggests that we can solve a square root equation by squaring both sides of the equation.

Extraneous solutions

Squaring both sides of an equation can, however, introduce extraneous solutions. Consider the equation

x = 6

The solution is 6. Square both sides.

x 2 = ( 6 ) 2
x 2 = 36

This equation has two solutions, 6 and + 6. The + 6 is an extraneous solution since it does not check in the original equation: + 6 6.

Method for solving square root equations

Solving square root equations

  1. Isolate a radical. This means get a square root expression by itself on one side of the equal sign.
  2. Square both sides of the equation.
  3. Simplify the equation by combining like terms.
  4. Repeat step 1 if radicals are still present.
  5. Obtain potential solutions by solving the resulting non-square root equation.
  6. Check each potential solution by substitution into the original equation.

Sample set a

Solve each square root equation.

x = 8. The radical is isolated Square both sides . ( x ) 2 = 8 2 x = 64 Check this potential solution . C h e c k : 64 = 8 Is this correct? 8 = 8 Yes, this is correct. 64 is the solution .

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y - 3 = 4. The radical is isolated. Square both sides . y - 3 = 16 Solve this nonradical equation . y = 19 Check this potential solution . C h e c k : 19 - 3 = 16 Is this correct? 16 = 4 Is this correct? 4 = 4 Yes, this is correct. 19 is the solution .

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2 m + 3 - m - 8 = 0. Isolate either radical . 2 m + 3 = m - 8 Square both sides . 2 m + 3 = m - 8 Solve this nonradical equation . m = - 11 Check this potential solution . C h e c k : 2 ( - 11 ) + 3 - ( - 11 ) - 8 = 0 Is this correct? - 22 + 3 - - 19 = 0 Is this correct?
Since  - 19  is not a real number, the potential solution of  m = - 11  does not check . This equation has no real solution .

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4 x 5 = 6. By inspection, this equation has no real solution .
The symbol, , signifies the positive square root and not the negative square root.

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Practice set a

Solve each square root equation.

3 a + 8 2 a + 5 = 0

a = 3 is extraneous, no real solution

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m 4 = 11

no real solution

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Exercises

For the following problems, solve the square root equations.

y 4 4 = 0

y = 20

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6 m 4 = 5 m 1

m = 3

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7 a + 6 = 3 a 18

no solution

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10 a 7 2 a + 9 = 0

a = 2

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12 k 5 9 k + 10 = 0

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x 6 3 x 8 = 0

no solution

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4 a 5 7 a 20 = 0

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2 m 6 = m 2

m = 4

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3 x + 1 = 2 x 6

no solution

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2 a + 9 a 4 = 0

no solution

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At a certain electronics company, the daily output Q is related to the number of people A on the assembly line by Q = 400 + 10 A + 125 .
(a) Determine the daily output if there are 44 people on the assembly line.

(b) Determine how many people are needed on the assembly line if the daily output is to be 520.

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At a store, the daily number of sales S is approximately related to the number of employees E by S = 100 + 15 E + 6
(a) Determine the approximate number of sales if there are 19 employees.

(b) Determine the number of employees the store would need to produce 310 sales.

( a ) S = 175 ; ( b ) E = 190

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Use a calculator. The resonance frequency f in an electronic circuit containing inductance L and capacitance C in series is given by

f = 1 2 π L C

(a) Determine the resonance frequency in an electronic circuit if the inductance is 4 and the capacitance is 0.0001 . Use π = 3.14.

(b) Determine the inductance in an electric circuit if the resonance frequency is 7.12 and the capacitance is 0.0001 . Use π = 3.14.

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If two magnetic poles of strength m and m ' units are at a distance r centimeters (cm) apart, the force F of repulsion in air between them is given by

F = m m ' r 2

(a) Determine the force of repulsion if two magnetic poles of strengths 20 and 40 units are 5 cm apart in air.

(b) Determine how far apart are two magnetic poles of strengths 30 and 40 units if the force of repulsion in air between them is 0.0001 .

( a ) F = 32 ( b ) r = 8 cm

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The velocity V in feet per second of outflow of a liquid from an orifice is given by V = 8 h , where h is the height in feet of the liquid above the opening.
(a) Determine the velocity of outflow of a liquid from an orifice that is 9 feet below the top surface of a liquid ( V is in feet/sec).

(b) Determine how high a liquid is above an orifice if the velocity of outflow is 81 feet/second.

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Use a calculator. The period T in seconds of a simple pendulum of length L in feet is given by T = 2 π L 32 .

(a) Determine the period of a simple pendulum that is 2 feet long. Use π = 3.14.

(b) Determine the length in feet of a simple pendulum whose period is 10.8772 seconds. Use π = 3.14.

( a ) T = 1.57  sec  ( b ) L = 95.99  cm

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The kinetic energy K E in foot pounds of a body of mass m in slugs moving with a velocity v in feet/sec is given by

K E = 1 2 m v 2

(a) Determine the kinetic energy of a 2-slug body moving with a velocity of 4 ft/sec.

(b) Determine the velocity in feet/sec of a 4-slug body if its kinetic energy is 50 foot pounds.

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Exercises for review

( [link] ) Write x 10 y 3 ( x + 7 ) 4 x 2 y 3 ( x + 7 ) 1 so that only positive exponents appear.

x 12 ( x + 7 ) 5

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( [link] ) Classify x + 4 = x + 7 as an identity, a contradiction, or a conditional equation.

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( [link] ) Supply the missing words. In the coordinate plane, lines with slope rise and lines with slope fall.

positive; negative

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( [link] ) Simplify ( x + 3 ) 4 ( x 2 ) 6 .

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( [link] ) Simplify ( 3 + 5 ) ( 4 5 ) .

7 + 5

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Questions & Answers

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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