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The idea behind solving equations containing square roots is to raise to power 3 in order to clear the cube root using the property
( cube_root( x ) ) 3 = x.
Example 1 : Find all real solutions to the equation
cube_root( x ) - x = 0
Solution to Example 1:
- Rewrite equation with the term containing cube root isolated
cube_root( x ) = x
- Raise both sides to power 3 in order to clear the cube root.
[ cube_root( x ) ] 3 = x 3
- Rewrite the above equation with right side equal to zero.
x - x 3 = 0
- Factor
x (1 - x 2) = 0
- and solve for x.
solutions are : x = 0 , x = - 1 and x = 1.
It is good to check the solutions found.
1. x = 0
Left side (LS) of the given equation when x = 0
LS = cube_root( x ) - x = cube_root (0) - 0 = 0
Right Side (RS) of the given equation when x = 0
RS = 0
2. x = -1
Left side (LS) of the given equation when x = -1
LS = cube_root( x ) - x = cube_root (-1) - (-1) = -1 + 1 = 0
Right Side (RS) of the given equation when x = -1
RS = 0
3. x = 1
Left side (LS) of the given equation when x = 1
LS = cube_root( x ) - x = cube_root (1) - 1 = 0
Right Side (RS) of the given equation when x = 1
RS = 0
Example 2 : Find all real solutions to the equation
cube_root( x 2 + 2 x + 8 ) = 2
Solution to Example 2:
- Given
cube_root( x 2 + 2 x + 8 ) = 2
- We raise both sides to power 3 in order to clear the cube root.
[ cube_root( x 2 + 2 x + 8 ) ] 3 = 2 3
- and simplify.
x 2 + 2 x + 8 = 8
- Rewrite the above equation with right side equal to zero.
x 2 + 2 x = 0
- Factor
x (x + 2) = 0
- and solve for x.
x = 0 and x = - 2.
Let us check the solutions obtained as an exercise.
1. x = 0
Left side (LS) of the given equation when x = 0
LS = cube_root( x 2 + 2 x + 8 ) = cube_root (0 + 0 + 8) = 2
Right Side (RS) of the given equation when x = 0
RS = 2
2. x = -2
Left side (LS) of the given equation when x = 0
LS = cube_root( x 2 + 2 x + 8 )
= cube_root ( (-2) 2 + 2*(-2) + 8 ) = cube_root ( 8 ) = 2
Right Side (RS) of the given equation when x = 0
RS = 2
Exercises:(answers further down the page)
Solve the following equations
1. cube_root( x ) - 4 x = 0
2. cube_root( x 2 + 2 x + 61 ) = 4
Solutions to above exercises
1. x = 0 , x = 1 / 8 , x = - 1 / 8
2. x = 1 , x = -3
More references and links on how to Solve Equations, Systems of Equations and Inequalities.
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