Irrational & Radical Equation Solver
Solve square root equations with step-by-step verification. Automatically identifies extraneous roots and validates domain constraints.
Solve: √x + =
About
Equations containing variables under a radical (square root) require specific algebraic manipulation to solve. The standard method involves isolating the radical and squaring both sides. However, this operation is irreversible and often introduces extraneous roots - values that satisfy the squared equation but not the original one. Checking the solution by substituting it back into the original equation is mandatory.
This tool automates the process for equations of the form √ax + b = c. It highlights the checking step, ensuring strict mathematical rigor.
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Formulas
To solve √U = V:
- Isolate the radical.
- Square both sides: U = V2.
- Solve for x.
- CHECK: Substitute x back into √U = V. If V < 0, the solution is invalid.
Reference Data
| Equation | Squared Form | Potential Roots | Valid Solution |
|---|---|---|---|
| √x = -3 | x = 9 | 9 | None (Extraneous) |
| √x + 5 = 4 | x + 5 = 16 | 11 | 11 |
| √2x = 4 | 2x = 16 | 8 | 8 |
Frequently Asked Questions
Squaring is not a one-to-one function. Both (-3)² and (3)² equal 9. If the original equation required the square root to equal -3, squaring "hides" that impossibility, producing a solution that works algebraically for the squared version but fails the original check.
In the real number system, the square root of a negative number is undefined. This solver operates within the set of Real Numbers (R).