<< Chapter < Page 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 ± i 1 3 and ± 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: −14 , and −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

what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
yeah
Morosi
prime number?
Morosi
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan 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