Solve radical equations of the form n√(ax + b) = c instantly. This tool provides step-by-step logic and checks for extraneous solutions.
Solve: n√(ax + b) = c
Solution:
What is a Radical Equation?
A radical equation is an algebraic equation in which at least one variable is located underneath a radical symbol (root). The most common type is the square root equation, but they can also involve cube roots or higher-order roots. In mathematical terms, any equation containing an expression like √x, ³√x, or n√x is classified as a radical equation.
Solving these equations requires "undoing" the radical by raising both sides of the equation to the power of the radical's index. However, this process can sometimes introduce extraneous solutions—values that look like solutions but do not satisfy the original equation.
Visualization of y = √x. Notice the domain is restricted to x ≥ 0.
The Formula and Logic
To solve the equation n√(ax + b) = c, we follow the Power Property of Equality:
(n√(ax + b))n = cn
ax + b = cn
From here, the equation becomes a standard linear equation:
- Subtract b from both sides:
ax = cn - b - Divide by a:
x = (cn - b) / a
Practical Examples
Example 1: Square Root
Solve √(2x + 4) = 6.
- Square both sides: 2x + 4 = 36
- Subtract 4: 2x = 32
- Divide by 2: x = 16
- Check: √(2(16) + 4) = √36 = 6. (Correct!)
Example 2: Cube Root
Solve ³√(x - 5) = 2.
- Cube both sides: x - 5 = 2³ = 8
- Add 5: x = 13
- Check: ³√(13 - 5) = ³√8 = 2. (Correct!)
How to Use the Radical Equation Calculator
Follow these simple steps to get your answer:
- Index (n): Enter the root number. Use 2 for square roots, 3 for cube roots, etc.
- Coefficient (a): Enter the number multiplying 'x' inside the radical.
- Constant (b): Enter the number being added or subtracted inside the radical.
- Equality (c): Enter the value the radical is equal to.
- Solve: Click "Solve Equation" to see the result and the verification steps.
Key Factors to Remember
| Factor | Description | Impact |
|---|---|---|
| Index Parity | Whether the root index (n) is even or odd. | Even roots cannot have negative radicands in real numbers. |
| Extraneous Solutions | Solutions that emerge from squaring/cubing but fail the check. | Occurs often when the right side (c) is negative for even roots. |
| Radicand | The expression (ax + b) inside the radical. | Must be ≥ 0 if the index is even. |
Frequently Asked Questions
What is an extraneous solution?
An extraneous solution is a root of a transformed equation (like after squaring both sides) that is not a root of the original equation.
Can a square root equal a negative number?
In the real number system, the principal square root is always non-negative. Therefore, √x = -5 has no real solution.
How do you isolate a radical?
Move all terms not inside the radical to the opposite side of the equation using addition or subtraction.
What if there are two radicals?
Isolate one radical, square both sides, then isolate the remaining radical and square again.
Is the index always 2?
No, the index can be any positive integer. If no index is shown, it is assumed to be 2 (square root).
Does this calculator handle complex numbers?
This specific version focuses on real number solutions, which is the standard for most algebra curricula.
Why do we check solutions?
Because raising an equation to an even power can turn a negative value into a positive one, potentially creating "fake" answers.
Can 'a' be zero?
If 'a' is zero, the variable 'x' disappears, and you are simply checking if a constant root equals 'c'.