Problem 4 Identifying Extraneous Solutions

What is the solution of n = n + 12 ? n equals , square root of n plus 12 end root , question mark

n = n − 12   n 2 = ( n + 12 ) 2 Square each side . n 2 = n + 12 Simplify . n 2 − n − 12 = 0 Subtract  n + 12   from each side . ( n − 4 ) ( n + 3 ) = 0 Factor the quadratic equation . n − 4 = 0 or n + 3 = 0 Use the Zero-Product Property . n = 4 or n = − 3 Solve for  n . table with 7 rows and 3 columns , row1 column 1 , n , column 2 equals , square root of n minus 12 end root , column 3 , row2 column 1 , n squared , column 2 equals . open , square root of n plus 12 end root , close squared , column 3 cap squareeachside . . , row3 column 1 , n squared , column 2 equals n plus 12 , column 3 cap simplify , . , row4 column 1 , n squared , minus n minus 12 , column 2 equals 0 , column 3 cap subtract . n plus 12 . fromeachside . . , row5 column 1 , open , n minus 4 , close . open , n plus 3 , close , column 2 equals 0 , column 3 cap factorthequadraticequation . . , row6 column 1 , n minus 4 equals 0 , column 2 or n plus 3 equals 0 , column 3 cap usethecap zerominuscap productcap property . . , row7 column 1 , n equals 4 , column 2 or n equals negative 3 , column 3 cap solvefor . n . , end table

Check

Check 4 = ? 4 + 12 Substitute  4  and  − 3  for  n . − 3 = ? − 3 + 12 4 = 4 ✔ − 3 ≠ 6 table with 2 rows and 4 columns , row1 column 1 , cap check , 4 , column 2 modified equals with question mark above . square root of 4 plus 12 end root . cap substitute . 4 , and , minus 3 , for , n . , column 3 negative 3 , column 4 modified equals with question mark above . square root of negative 3 plus 12 end root , row2 column 1 , 4 , column 2 equals 4 u 2 , column 3 negative 3 , column 4 not equal to 6 , end table

The solution of the original equation is 4. The value − 3 negative 3  is an extraneous solution.

✓ Got It?

4. What is the solution of − y = y + 6 ? negative y equals , square root of y plus 6 end root , question mark

Problem 5 Identifying Equations With No Solution

What is the solution of 3 y + 8 = 2 ? square root of 3 y end root , plus 8 equals 2 question mark

3 y + 8 = 2   3 y = − 6 Subtract  8  from each side . 3 y = 36 Square each side . y = 12 Divide each side by  3. table with 4 rows and 3 columns , row1 column 1 , square root of 3 y end root , plus 8 , column 2 equals 2 , column 3 , row2 column 1 , square root of 3 y end root , column 2 equals negative 6 , column 3 cap subtract . 8 . fromeachside . . , row3 column 1 , 3 y , column 2 equals 36 , column 3 cap squareeachside . . , row4 column 1 , y , column 2 equals 12 , column 3 cap divideeachsideby . 3. , end table

Check 3 ( 12 ) + 8 = ? 2 Substitute  12  for  y . 14 ≠ 2 y = 12  does not satisfy the original equation . table with 2 rows and 2 columns , row1 column 1 , cap check . square root of 3 , open 12 close end root . plus 8 , column 2 modified equals with question mark above , 2 . cap substitute . 12 , for , y . , row2 column 1 , 14 , column 2 not equal to 2 y equals 12 . doesnotsatisfytheoriginalequation . . , end table

The apparent solution 12 is extraneous. The original equation has no solution.