The solving method

The standard method for an irrational equation with one square root isolates the radical, squares it and then checks the solutions.

In brief — Solving method

To solve an irrational equation with a square root:

1. Isolate the radical on one side.
2. Determine the domain of existence: argument $\ge 0$.
3. Determine the sign-agreement condition: the member without the root must be $\ge 0$.
4. Square both members.
5. Solve the resulting equation.
6. Check that the solutions satisfy the domain of existence and the sign-agreement condition (discard the extraneous solutions).

Remark — Compact scheme $\sqrt{A(x)}=B(x)$

Many irrational equations appear in the “isolated” form $\sqrt{A(x)}=B(x)$. In that case the equivalent system is: $\sqrt{A(x)}=B(x) \iff \begin{cases} B(x)\ge 0 \\ A(x) = B^2(x) \end{cases}$ The existence condition $A(x)\ge 0$ is optional: if $A(x)=B^2(x)$, then automatically $A(x)\ge 0$ because a square cannot be negative. It remains necessary, however, to write the domain of existence for all the other roots that may be present in the equation.

Links

Topics: Radicals
Concepts: Conditions of existence · Irrational equation · Extraneous solution
Methods: Irrational equation squaring
Skills: Solving equations