1. Home
  2. College algebra
  3. Equations and inequalities
  4. Other types of equations

2.6 Other types of equations  (Page 4/10)

<< Chapter < Page
  College algebra     Page 4 / 10
Chapter >> Page >

Solve the absolute value equation: | 1 − 4 x | + 8 = 13.

x = −1 , x = 3 2

Got questions? Get instant answers now!

Solving other types of equations

There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.

Solving equations in quadratic form

Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include   x 4 − 5 x 2 + 4 = 0 , x 6 + 7 x 3 − 8 = 0 , and   x 2 3 + 4 x 1 3 + 2 = 0.   In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.

Quadratic form

If the exponent on the middle term is one-half of the exponent on the leading term, we have an equation in quadratic form , which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.

Given an equation quadratic in form, solve it.

  1. Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.
  2. If it is, substitute a variable, such as u , for the variable portion of the middle term.
  3. Rewrite the equation so that it takes on the standard form of a quadratic.
  4. Solve using one of the usual methods for solving a quadratic.
  5. Replace the substitution variable with the original term.
  6. Solve the remaining equation.

Solving a fourth-degree equation in quadratic form

Solve this fourth-degree equation:   3 x 4 − 2 x 2 − 1 = 0.

This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let   u = x 2 . Rewrite the equation in u .

3 u 2 − 2 u − 1 = 0

Now solve the quadratic.

3 u 2 − 2 u − 1 = 0 ( 3 u + 1 ) ( u − 1 ) = 0

Solve each factor and replace the original term for u.

3 u + 1 = 0 3 u = −1 u = − 1 3 x 2 = − 1 3 x = ± i 1 3
u − 1 = 0 u = 1 x 2 = 1 x = ±1

The solutions are   x = ± i 1 3   and   x = ± 1.

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

Solve using substitution:   x 4 − 8 x 2 − 9 = 0.

x = −3 , 3 , − i , i

Got questions? Get instant answers now!

Solving an equation in quadratic form containing a binomial

Solve the equation in quadratic form:   ( x + 2 ) 2 + 11 ( x + 2 ) − 12 = 0.

This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting   u = x + 2.   Then rewrite the equation in u.

u 2 + 11 u − 12 = 0 ( u + 12 ) ( u − 1 ) = 0

Solve using the zero-factor property and then replace u with the original expression.

u + 12 = 0 u = −12 x + 2 = −12 x = −14

The second factor results in

u − 1 = 0 u = 1 x + 2 = 1 x = −1

We have two solutions:   x = −14 , x = −1.

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

Solve:   ( x − 5 ) 2 − 4 ( x − 5 ) − 21 = 0.

x = 2 , x = 12

Got questions? Get instant answers now!

Solving rational equations resulting in a quadratic

Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.

Questions & Answers

summarize halerambos & holbon
David Reply
the Three stages of Auguste Comte
Clementina Reply
what are agents of socialization
Antonio Reply
sociology of education
Nuhu Reply
definition of sociology of education
Nuhu
what is culture
Abdulrahim Reply
shared beliefs, values, and practices
AI-Robot
What are the two type of scientific method
ogunniran Reply
I'm willing to join you
Aceng Reply
what are the scientific method of sociology
Man
what is socialization
ogunniran Reply
the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values
AI-Robot
scientific method in doing research
ogunniran
defimition of sickness in afica
Anita
Cosmology
ogunniran
Hmmm
ogunniran
list and explain the terms that found in society
REMMY Reply
list and explain the terms that found in society
Mukhtar
what are the agents of socialization
Antonio
Family Peer group Institution
Abdulwajud
I mean the definition
Antonio
ways of perceived deviance indifferent society
Naomi Reply
reasons of joining groups
SAM
to bring development to the nation at large
Hyellafiya
entails of consultative and consensus building from others
Gadama
World first Sociologist?
Abu
What is evolutionary model
Muhammad Reply
Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u
faruk
what are the modern trends in religious behaviours
Selekeye Reply
what are social norms
Daniel Reply
shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society
Lucius
that is how i understood it
Lucius
examples of societal norms
Diamond
Discuss the characteristics of the research located within positivist and the interpretivist paradigm
Tariro Reply
what is Industrialisation
Selekeye Reply
industrialization
Angelo
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask