<< Chapter < Page Chapter >> Page >

Solving a rational equation leading to a quadratic

Solve the following rational equation: 4 x x 1 + 4 x + 1 = 8 x 2 1 .

We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However, x 2 −1 = ( x + 1 ) ( x 1 ) . Then, the LCD is ( x + 1 ) ( x 1 ) . Next, we multiply the whole equation by the LCD.

( x + 1 ) ( x 1 ) [ −4 x x 1 + 4 x + 1 ] = [ −8 ( x + 1 ) ( x 1 ) ] ( x + 1 ) ( x 1 ) −4 x ( x + 1 ) + 4 ( x 1 ) = −8 −4 x 2 4 x + 4 x 4 = −8 −4 x 2 + 4 = 0 −4 ( x 2 1 ) = 0 −4 ( x + 1 ) ( x 1 ) = 0 x = −1 x = 1

In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Solve 3 x + 2 x 2 + 1 x = 2 x 2 2 x .

x = −1 , x = 0 is not a solution.

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with different types of equations.

Key concepts

  • Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See [link] , [link] , and [link] .
  • Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See [link] and [link] .
  • We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See [link] and [link] .
  • To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See [link] .
  • Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See [link] and [link] .
  • Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See [link] .

Section exercises

Verbal

In a radical equation, what does it mean if a number is an extraneous solution?

This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.

Got questions? Get instant answers now!

Explain why possible solutions must be checked in radical equations.

Got questions? Get instant answers now!

Your friend tries to calculate the value 9 3 2 and keeps getting an ERROR message. What mistake is he or she probably making?

He or she is probably trying to enter negative 9, but taking the square root of −9 is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in −27.

Got questions? Get instant answers now!

Explain why | 2 x + 5 | = −7 has no solutions.

Got questions? Get instant answers now!

Explain how to change a rational exponent into the correct radical expression.

A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.

Got questions? Get instant answers now!

Algebraic

For the following exercises, solve the rational exponent equation. Use factoring where necessary.

( x 1 ) 3 4 = 8

x = 17

Got questions? Get instant answers now!

x 2 3 5 x 1 3 + 6 = 0

x = 8 ,     x = 27

Got questions? Get instant answers now!

x 7 3 3 x 4 3 4 x 1 3 = 0

Got questions? Get instant answers now!

For the following exercises, solve the following polynomial equations by grouping and factoring.

x 3 + 2 x 2 x 2 = 0

x = −2 , 1 , −1

Got questions? Get instant answers now!

3 x 3 6 x 2 27 x + 54 = 0

Got questions? Get instant answers now!

4 y 3 9 y = 0

y = 0 ,     3 2 ,     3 2

Got questions? Get instant answers now!

x 3 + 3 x 2 25 x 75 = 0

Got questions? Get instant answers now!

m 3 + m 2 m 1 = 0

m = 1 , −1

Got questions? Get instant answers now!

5 x 3 + 45 x = 2 x 2 + 18

x = 2 5

Got questions? Get instant answers now!

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

3 x + 7 + x + 2 = 1

x = −2

Got questions? Get instant answers now!

For the following exercises, solve the equation involving absolute value.

| 3 x 4 | = 8

x = 4 , −4 3

Got questions? Get instant answers now!

| 1 4 x | 1 = 5

x = 5 4 , 7 4

Got questions? Get instant answers now!

| 2 x 1 | 7 = −2

x = 3 , −2

Got questions? Get instant answers now!

| 2 x + 1 | 2 = −3

Got questions? Get instant answers now!

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

x 4 10 x 2 + 9 = 0

x = 1 , −1 , 3 , −3

Got questions? Get instant answers now!

4 ( t 1 ) 2 9 ( t 1 ) = −2

Got questions? Get instant answers now!

( x 2 1 ) 2 + ( x 2 1 ) 12 = 0

x = 2 , −2

Got questions? Get instant answers now!

( x + 1 ) 2 8 ( x + 1 ) 9 = 0

Got questions? Get instant answers now!

( x 3 ) 2 4 = 0

x = 1 , 5

Got questions? Get instant answers now!

Extensions

For the following exercises, solve for the unknown variable.

x −2 x −1 12 = 0

Got questions? Get instant answers now!

| x | 2 = x

All real numbers

Got questions? Get instant answers now!

| x 2 + 2 x 36 | = 12

x = 4 , 6 , −6 , −8

Got questions? Get instant answers now!

Real-world applications

For the following exercises, use the model for the period of a pendulum, T , such that T = 2 π L g , where the length of the pendulum is L and the acceleration due to gravity is g .

If the acceleration due to gravity is 9.8 m/s 2 and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).

Got questions? Get instant answers now!

If the gravity is 32 ft/s 2 and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.

10 in.

Got questions? Get instant answers now!

For the following exercises, use a model for body surface area, BSA, such that B S A = w h 3600 , where w = weight in kg and h = height in cm.

Find the height of a 72-kg female to the nearest cm whose B S A = 1.8.

Got questions? Get instant answers now!

Find the weight of a 177-cm male to the nearest kg whose B S A = 2.1.

90 kg

Got questions? Get instant answers now!

Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
cos(- z)=cos z
Mustafa
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 5

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask