<< 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

(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
modul questions trigonometry
Thamarai Reply
(1+cosA)(1-cosA)=sin^2A
BINCY
differentiate f(t)=1/4t to the power 4 +8
Jessica Reply
I need trigonometry,polynomial
duru Reply
ok
Augustine
Why is 7 on top
Bertha Reply
simplify cot x / csc x
Catherine Reply
👉🌹Solve🌻 Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
what is the period of cos?
SIYAMTHEMBA Reply
your question might not seem clear as you asked. ask well to get perfect answers put your question on a table I'm willing to help you Mr Siyamthemba
Patrick
simplify: cot x/csc x
Catherine
sorry i didnt realize you were actually asking someone else to put their question on here. i thought this was where i was supposed to.
Catherine
some to dereve formula for bulky density
kurash
Solve Given that: cotx/cosx =cosx/sinx/cosx =1/sinx =cosecx Ans.
Vijay
if tan alpha + beta is equal to sin x + Y then prove that X square + Y square - 2 I got hyperbole 2 Beta + 1 is equal to zero
Rahul Reply
questions
Thamarai
ok
AjA
sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
i don't need help solving it I just need a memory jogger please.
Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
Mintah Reply
its a very simple
Kavita
gave your expression then i solve
Kavita
Hy guys, I have a problem when it comes on solving equations and expressions, can you help me 😭😭
Thuli
Tomorrow its an revision on factorising and Simplifying...
Thuli
ok sent the quiz
kurash
send
Kavita
Hi
Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
Masum
no I m right
Kavita
No sister.
Masum
no I m right
Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah 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