Radical Equations Checker
Square both sides, solve the resulting polynomial, then verify each candidate in the original equation.
Algebraic Domain Invariants 🖖
Squaring a radical equation intentionally destroys the parity of the analytical domain, inherently conjuring extraneous phantom roots. These mathematical artifacts perfectly satisfy the quadratic formulation but violently violate the domain restrictions of the principal square root.
Why you must check every answer 🖖
Squaring both sides is a one-way street: because both 3 and −3 square to 9, that step can admit answers the original equation never allowed. The root symbol returns only the non-negative value, so the right side cx+d must be ≥ 0. Takeaway: substitute every candidate back into the untouched original equation — this tool does it automatically and flags the impostors.
Extraneous roots solve the mirror equation 🖖
A rejected root is not meaningless — it is the exact solution of the sign-flipped twin √(ax+b) = −(cx+d). Squaring erases the minus sign, so both equations share one polynomial, and its candidates split cleanly: those with cx+d ≥ 0 belong to the original, the rest to its mirror. That is why every extraneous point lands precisely on the dashed negative branch y = −√(ax+b) in the plot.
Example problems
- single-root case - Single valid solution case for sqrt(x+4)=x.
- extraneous root case - Squaring introduces a candidate that fails original-domain check.
- two-candidate case - Two algebraic candidates appear before radical-domain filtering.