Equation

Published on November 2016 | Categories: Documents | Downloads: 82 | Comments: 0 | Views: 539
of 3
Download PDF   Embed   Report

Finding all real solution to the given equation

Comments

Content



1

Show complete solutions and box the final answers.

Find all real solutions to the equation
  1 1
8 2
2
2
  
  x x
x x


(Hint: The solution set has exactly 5 elements. If a,bεR, when will a
b
be equal to 1?)

Solution:

Let
  1
2
   x x a
---base
8 2
2
 

x x
b
---exponent

There are three possible ways that make the equation equal to 1.

In accord with the laws of indices/exponent, the equation is equal to 1 if:

1. In the expression given (a
b
), b =0 & a is any integer (except 0, because 0
0
is indeterminate)
Recall: x
0
=1

Finding the real solution
a. Equate 8 2
2
  x x to 0, that is
0 8 2
2
   x x
b. Factor the polynomial
   2 4   x x
c. Equate both factors to 0 and find the values of x, that is
0 4   x 0 2   x
4  x   2   x

x = 4 , x = -2




2. In the expression given(a
b
), a is 1 & b is any integer
Recall: 1
n
= 1

Finding the real solution

2

a. Equate 1
2
  x x to one, that is
1 1
2
   x x
Transpose 1 to the left, thus becomes:
0 2
2
   x x
b. Factor the Polynomial
   2 1   x x
c. Equate both factors to 0 and find the values of x, that is
  0 1   x   0 2   x
1   x 2  x

x = -1 , x = 2



3. In the expression given(a
b
), a is -1 & b is even integer
Recall: (-1)
n
= 1, where n is an even integer

Finding the real solution
a. Find the values of x for which 1 1
2
    x x
1 1
2
    x x
0
2
  x x
By factoring:    0 1   x x
0  x ; 0  x
0 1  x ; 1  x


b. For each solution, let’s see if the exponent is an even integer by substituting the values of x
Let 0  x
8 2
2
  x x ; 8 ) 0 ( 2 0
2
  8 0 0   8 8 0    (EVEN)
Let 1  x
8 2
2
  x x ; 8 ) 1 ( 2 1
2
  8 2 1   9 10 1    (ODD)


Thus, the value of x that makes the exponent equal to an even integer is 0.

x = 0


TABLE OF SUMMARY

3


VALUE OF a VALUE OF b REAL SOLUTIONS (x) RESULT
any integer,
except 0
0 4 , -2 1
1 any integer -1 , 2 1
-1 even integer 0 1









Sponsor Documents

Or use your account on DocShare.tips

Hide

Forgot your password?

Or register your new account on DocShare.tips

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close