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

x = −1 , x = 3 2

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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 ± i 1 3 and ± 1.

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

x = −3 , 3 , i , i

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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: −14 , and −1.

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

x = 2 , x = 12

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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

what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
comment écrire les symboles de math par un clavier normal
SLIMANE
Practice Key Terms 5

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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