**Radical** **equations** (also known as **irrational**) are equations in which the unknown value appears under a radical sign.

The method for solving radical equation is raising both sides of the equation to the same power.

If we have the equation , then the condition of that equation is always , however, this is not a sufficient condition. Respecting the properties of the square root function (the domain of square root function is ), the second condition is . Therefore, we need to ensure that both sides of equation are non-negative.

After squaring we have an equivalent equation:

Condition is now unnecessary (it is automatically satisfied after squaring); the solutions of the equation will thus satisfy condition , so that for these solutions it will be .

Now we have:

and .

In general, this is valid for the square root of every even number :

and .

For the square root of every odd number it will be

,

because their domain is a whole set of real numbers.

*Example 1.* Solve the equation:

.

*Solution*:

Both sides of the equation are non-negative; we can square the equation:

.

We must now confirm if it is the correct solution:

.

It follows that is the solution of the given equation.

*Example 2*. Solve the equation:

.

*Solution*:

Conditions for this equation are and and .

It follows that must be in interval .

Both sides of the equation are always non-negative, therefore we can square the given equation.

.

We need check that is the solution of the initial equation:

.

It follows that is the solution of the initial equation. We can conclude that directly from the condition of the equation, without any further requirement to checking.

*Example 3*. Solve the equation:

.

*Solution*:

Now we must be sure that the right side of the equation is non-negative. Therefore is the condition of this equation.

After squaring the equation, we have:

.

The solutions for quadratic equation are:

and .

The only solution is due to satisfied condition .

*Example 4*. Solve the equation:

*Solution*:

Both sides of the equation are always non-negative, therefore we can square the equation. In this example we need to square the equation twice, as displayed below:

.

is not the solution of the initial equation, because , which is the condition of the equation (check it!).

Note: as we observed through the steps of solving of the equation, that this equation does not have solutions before the second squaring, because the square root cannot be negative.

*Example 5.* Solve the equation.

.

*Solution*:

Both sides of the equation are non-negative, therefore we can square the equation:

.

Let’s check that satisfies the initial equation:

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It follows that is the solution of the given equation.

## Radical equations worksheets

**Solve radical equations** (370.6 KiB, 180 hits)