Radical Equations

Radical equations are mathematical equations in which the variable appears inside a radical, such as a square root, cube root, or fourth root.

Solving them involves eliminating the radical by raising both sides of the equation to a power equal to the index of the radical.

For instance, calculating the depth of a well by dropping an object and measuring the time until the splash is heard is a practical application of a radical equation.

Here, total time includes the fall time, given by the square root of twice the depth over gravitational acceleration, and the sound travel time, given by the depth over the speed of sound, forming a radical equation.

The first step is isolating the radical term and then squaring both sides to eliminate the square root.

This step changes the radical equation into a standard quadratic equation, which can be simplified further. The quadratic formula is applied to solve it, giving two possible values for the depth.

Each solution is substituted back into the original radical equation to check if it truly satisfies the equation or if it is an extraneous root.

Only one solution provides a valid, positive number for depth, confirming it as the correct solution.