Problem 5 Solving an Equation With Two Radicals

What is the solution of 2 x + 1 − x = 1 ? square root of 2 x plus 1 end root . minus square root of x equals 1 question mark

2 x + 1 − x = 1   2 x + 1 = x + 1 Isolate the more complicated radical . ( 2 x + 1 ) 2 = ( x + 1 ) 2 Square each side . 2 x + 1 = x + 2 x + 1   x = 2 x Isolate  2 x . x 2 = ( 2 x ) 2 Square each side . x 2 = 4 x   x 2 − 4 x = 0 Subtract  4 x  from each side . x ( x − 4 ) = 0 Factor . x = 0  or  x = 4 Zero-Product Property table with 10 rows and 3 columns , row1 column 1 , square root of 2 x plus 1 end root . minus square root of x , column 2 equals 1 , column 3 , row2 column 1 , square root of 2 x plus 1 end root , column 2 equals square root of x plus 1 , column 3 cap isolatethemorecomplicatedradical . . , row3 column 1 , open . square root of 2 x plus 1 end root . close squared , column 2 equals . open , square root of x plus 1 , close squared , column 3 cap squareeachside . . , row4 column 1 , 2 x plus 1 , column 2 equals x plus 2 square root of x plus 1 , column 3 , row5 column 1 , x , column 2 equals 2 square root of x , column 3 cap isolate , 2 square root of x . , row6 column 1 , x squared , column 2 equals . open , 2 square root of x , close squared , column 3 cap squareeachside . . , row7 column 1 , x squared , column 2 equals 4 x , column 3 , row8 column 1 , x squared , minus 4 x , column 2 equals 0 , column 3 cap subtract . 4 x . fromeachside . . , row9 column 1 , x . open , x minus 4 , close , column 2 equals 0 , column 3 cap factor , . , row10 column 1 , x , column 2 equals 0 , or , x equals 4 , column 3 cap zerominuscap productcap property , end table

Check

2 x + 1 − x = 1 2 ( 0 ) + 1 − 0 = ? 1 1 − 0 = ? 1 1 − 0 = ? 1 1 = 1 ✓ 2 x + 1 − x = 1 2 ( 4 ) + 1 − 4 = ? 1 9 − 4 = ? 1 3 − 2 = ? 1 1 = 1 ✓ table with 1 row and 2 columns , row1 column 1 , table with 5 rows and 2 columns , row1 column 1 , square root of 2 x plus 1 end root . minus square root of x , column 2 equals 1 , row2 column 1 , square root of 2 , open 0 close , plus 1 end root . minus square root of 0 , column 2 modified equals with question mark above , 1 , row3 column 1 , square root of 1 minus 0 , column 2 modified equals with question mark above , 1 , row4 column 1 , 1 minus 0 , column 2 modified equals with question mark above , 1 , row5 column 1 , 1 , column 2 equals 1 check mark , end table , column 2 table with 5 rows and 2 columns , row1 column 1 , square root of 2 x plus 1 end root . minus square root of x , column 2 equals 1 , row2 column 1 , square root of 2 , open 4 close , plus 1 end root . minus square root of 4 , column 2 modified equals with question mark above , 1 , row3 column 1 , square root of 9 minus square root of 4 , column 2 modified equals with question mark above , 1 , row4 column 1 , 3 minus 2 , column 2 modified equals with question mark above , 1 , row5 column 1 , 1 , column 2 equals 1 check mark , end table , end table

The solutions are 0 and 4.

✓ Got It?

5. What is the solution of 5 x + 4 − x = 4 ? square root of 5 x plus 4 end root . minus square root of x equals 4 question mark

✓ Lesson Check

Do you know HOW?

Solve. Check for extraneous solutions.

1. 4 x − 23 − 3 = 2 square root of 4 x minus 23 end root . minus 3 equals 2
2. − x 3 + 3 = 0 negative , cube root of x , , plus 3 equals 0
3. 5 x + 7 = 8 5 square root of x plus 7 equals 8
4. 3 x = 6 3 square root of x equals 6
5. 5 − 2 x = 3 5 minus 2 square root of x equals 3
6. x 3 = 8 cube root of x , , equals 8

Do you UNDERSTAND?

7. Vocabulary Which value, 12 or 3, is an extraneous solution of x − 6 = 3 x ? x minus 6 equals , square root of 3 x end root , question mark  Explain your reasoning.
8. Compare and Contrast How is solving a square root equation similar to solving an absolute value equation? How is it different?