Solving Radical Equations

We examine ways to solve equations such as .

In solving any equation, we always “undo” the operation so that we can get x isolated. In this case we have to square both sides of the equation. The examples cited below describe some of the methods needed to solve equations which include radicals.

#1. To solve

we first square both sides of the equation.

The result is x - 2 = 81. This equation is simple to solve. We have x = 83.

#2. A more complicated situation is .

In this case we still begin by squaring both sides of the equation. The result is .

To finish solving this requires us to set all terms equal to zero and either factor or use the quadratic formula. We get .

This factors:

and the solutions are x = 2 or x = - 1.

We must check each of these solutions in the original equation to see if the value of x gives a solution.

x = 2 gives

which is correct.

x = - 1 gives

which is impossible since

for n > 0 is always positive.

Therefore the only solution is x = 2.

Keeping in mind the basic idea of squaring both sides of the equation in order to solve for x when there is a radical in the equation, we will now show some additional examples.

#3. Solve for x if .

Squaring both sides of the equation gives us .

Setting terms equal to zero gives

.

The expression factors:

.

Setting each factor equal to 0 gives two possible answers: x = 3 or x = – 1.

We check each answer in the original equation:

If x = 3 we have

which is correct.

If x = – 1 we have

which is impossible since the square root cannot be negative.

Therefore the only answer is x = 3.

#4. Suppose

.

We begin again by squaring both sides of the equation. The result is 3x – 2 = x + 7.

We solve for x getting 2x = 9 or .

We check this possible answer in the original equation.

If

we have .

This is the same as

which is correct.

Therefore the solution is .

#5. Solve for x if .

We square both sides. This gives x + 2 = 2 – x.

Solving for x gives 2x = 0. The only solution is x = 0.

Checking this in the original equation gives

which is correct.

Therefore the solution is x = 0.

#6. Let .

Our first step is to separate the radicals. The result is

.

Now we square both sides. The result is

.

We next isolate the radical term on the right side. The result is

.

We square both sides again. The result is .

We set the terms equal to zero. The result is

.

We solve this by factoring or using the quadratic formula. This does factor: (x – 5)(x + 3) = 0. This gives two possible solutions: x = 5 or x = – 3.

We check these possible solutions in the original equation.

If x = 5 we have . This is the same as 4 – 3 = 1.

If x = – 3 we have .

This is impossible since

.

Therefore the only solution is x = 5.

Another way to obtain the same result in example #6 is by using the Intersect feature on a graphing calculator. The graphs of Y1=

and Y2 = 1 are shown below intersecting where x = 5.