SOLVING RADICAL EQUATIONS
Radical equation is the one in which the variable appears inside the radical sign.
The following steps will be useful to solve a radical equation.
Step 1 :
Get rid of the radical sign by raising both sides of the equation to the index of the radical.
For example, to get rid of a square root, raise both sides of the equation to the power 2 and to get rid of a cube root, raise both sides of the equation to the power 3.
Step 2 :
Once the radical is completely removed, solve for the variable.
Step 3 :
After having solved for the variable, check the solution (value/s of the variable) with the original equation. If the value of the variable satisfies the original equation, it can be considered as a solution to the equation, otherwise not.
Example 1 :
Solve for x :
√(x + 1) + 7 = 10
Solution :
√(x + 1) + 7 = 10
Subtract 7 from both sides.
√(x + 1) = 3
Raise both sides to the power 2.
[√(x + 1)]2 = 32
x + 1 = 9
Subtract 1 from both sides.
x = 8
Substitute x = 8.
√(8 + 1) + 7 = 10
√9 + 7 = 10
3 + 7 = 10
10 = 10 (true)
So, 8 is the solution for the given radical equation.
Example 2 :
Solve for x :
√(x + 2) = x - 4
Solution :
√(x + 2) = x - 4
Raise both sides to the power 2.
[√(x + 2)]2 = (x - 4)2
x + 2 = (x - 4)(x - 4)
x + 2 = x2 - 4x - 4x + 16
x + 2 = x2 - 8x + 16
Subtract x and 2 from both sides.
0 = x2 - 9x + 14
Factor and solve :
(x - 2)(x - 7) = 0
x -2 = 0 or x - 7 = 0
|
x - 2 = 0 x = 2 |
x - 7 = 0 x = 7 |
Substitute x = 2 and x = 7 in the given equation.
|
√(x + 2) = x - 4 Substitute x = 2. √(2 + 2) = 2 - 4 √4 = -2 2 = -2 (false) |
√(x + 2) = x - 4 Substitute x = 7. √(7 + 2) = 7 - 4 √9 = 3 3 = 3 (true) |
x = 2 does not satisfy the given equation
So, x = 7 is the only solution.
Example 3 :
Solve for x :
√(3x - 5) = x - 5
Solution :
√(3x - 5) = x - 5
Raise both sides to the power 2.
[√(3x - 5)]2 = (x - 5)2
3x - 5 = (x - 5)(x - 5)
3x - 5 = x2 - 5x - 5x + 25
3x - 5 = x2 - 10x + 25
Subtract 3x from both sides and 5 to both sides.
0 = x2 - 13x + 30
Factor and solve :
(x - 10)(x - 3) = 0
x - 10 = 0 or x - 3 = 0
|
x - 10 = 0 x = 10 |
x - 3 = 0 x = 3 |
Substitute x = 10 and x = 3 in the given equation.
|
√(3x - 5) = x - 5 Substitute x = 10. √30 - 5 = 10 - 5 √25 = 5 5 = 5 (false) |
√(3x - 5) = x - 5 Substitute x = 3. √9 - 5 = 3 - 5 √4 = -2 2 = -2 (false) |
x = 3 does not satisfy the original equation.
So, 10 is the only solution.
Example 4 :
Check whether the following radical equation has solution or not.
√(x - 4) - √x = 2
Solution :
√(x - 4) - √x = 2
Add √x to both sides.
√(x - 4) = 2 + √x
Raise both sides to the power 2.
[√(x - 4)2 = [2 + √(x)]2
Using the identity (a - b)2 = a2 - 2ab + b2 on the right side,
[√(x - 4)]2 = 22 + 2(2)√x + [√x]2
x - 4 = 4 + 4√x + x
x - 4 = 4 + 4√x + x
Subtract x from both sides.
-4 = 4 + 4√x
Subtract 4 from both sides.
-8 = 4√x
Divide both sides by 4.
-2 = √x
Raise both sides to the power 2.
(-2)2 = (√x)2
4 = x
Substitute x = 4 in the given equation.
√(x - 4) - √x = 2
Substitute x = 4.
√(4 - 4) - √4 = 2
0 - 2 = 2
-2 = 2 (false)
So, the given equation has no solution.
Example 5 :
Check whether the following radical equation has solution or not.
√x - √(x - 4) = 2
Solution :
√x - √(x - 4) = 2
Subtract √x from both sides.
-√(x - 4) = 2 - √x
Multiply both sides by -1.
-1[-√(x - 4)] = -1(2 - √x)
√(x - 4) = -2 + √x
√(x - 4) = √x - 2
Raise both sides to the power 2.
[√(x - 4)]2 = (√x - 2)2
Using the identity (a - b)2 = a2 - 2ab + b2 on the right side,
x - 4 = [√x]2 - 2(2)√x + 22
x - 4 = x - 4√x + 4
Subtract x from both sides.
-4 = -4√x + 4
Subtract 4 from both sides.
-8 = -4√x
Divide both sides by -4.
2 = √x
Raise both sides to the power 2.
22 = (√x)2
4 = x
Substitute x = 4 in the given equation.
√(x - 4) - √x = 2
Substitute x = 4.
√4 - √(4 - 4) = 2
2 - 0 = 2
2 = 2 (true)
The given radical equation has solution and it is 2.
Example 6 :
Solve for x :
3√(3x) - 9 = 0
Solution :
3√(3x) - 9 = 0
Add 9 to both sides.
3√(3x) = 9
Raise both sides to the power 3.
[3√(3x)]3 = 93
3x = 729
Divide both sides by 3.
x = 243
Substitute x = 243.
3√(3 ⋅ 243) - 9 = 0
3√729 - 9 = 0
9 - 9 = 0
0 = 0 (true)
So, 243 is the solution for the given radical equation.
Example 7 :
Solve for x :
3√(x + 2) + 7 = 0
Solution :
3√(x + 2) + 7 = 0
Subtract 7 from both sides.
3√(x + 2) = -7
Raise both sides to the power 3.
[3√(x + 2)]3 = (-7)3
x + 2 = -343
Subtract 2 from both sides.
x = -345
Substitute x = 4.
3√(-345 + 2) + 7 = 0
3√(-343) + 7 = 0
-7 + 7 = 0
0 = 0 (true)
So, -345 is the solution for the given radical equation.
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