How to solve equations with radicals. Solved exercises

Now I’m going to explain how to solve the equations with radicals, with exercises solved step by step.

Radical equations

What are the equations with radicals?

The equations with radicals are those where x is within a square root.

A priori, these equations are neither first nor second degree, depending on the rest of the terms of the equation.

Let’s see what is the procedure to solve them and a few examples of equations with radicals.

How to solve an equation with radicals

The procedure for solving equations with radicals is as follows:

  1. Leave the term with the radical only in one member of the equation
  2. Square both members (this way we eliminate the root)
  3. Operate and generally, we will obtain a second degree equation that we will have to solve
  4. Determine which solution is the right one and discard the other, since it is a virtual solution (we have forced it by squaring the members, but it does not exist in reality)

Solved exercise of equations with radicals step by step

I am going to explain to you now more slowly how to solve the equations with radicals, following the previous procedure while we solve some exercises.

For example:

radical equations

We start by leaving the root alone in one of the limbs:

equations with radical exercises solved

We squared both limbs:

equations with radical exercises solved step by step

In the first member, the root is annulled with the square and in the second member there remains a remarkable product to be developed:

How to solve equations with radicals

We are left with a second-degree equation. To solve it, we pass all the terms to the first member, leaving zero in the second member:

radical equations solved exercises

We solve the equation with the general formula to solve second degree equations:

Solved exercises of radical equations

equations with radical exercises

We obtain two solutions, which are:

solving equations with radicals

how to solve step-by-step radical equations

The two solutions are not valid, since one of them was created by us squaring both members, a square that did not exist in the original equation.

To find which solution is valid, we have to substitute the value of both solutions in the original equation and check which of them meets the equality.

For x=4:

second degree equations with solved radical exercises

Radical Equation Exercises

The iguladadad is fulfilled

For x=1

solve radical equations

equations with roots

Equality is not fulfilled.

Therefore, the solution of the equation is x=4.

Let’s see another example of an equation with radicals with decimal numbers. We are used to the fact that both the numbers and the solutions of the equations are integers, but it doesn’t have to be that way.

Let’s see it. For example:

radical equations

We leave the root term only on one of the members:

Equation with Resolved Radical Exercises

We squared both members:

examples of radical equations

I remind you that we have added this step, that is, we are adding a solution that does not really exist.

We develop the squares in both the first and the second member:

how to solve an equation with radicals

To arrive at this result we must have clear properties of the powers, as well as notable products.

We have a complete second degree equation. To solve it, we reorder terms and pass them all to a member of the equation to equal zero:

how to solve equations with roots

We solve it and we have these solutions:
Exercises - Radical Equations
We must check which of the 2 is the good one and therefore, which one we discard. For them we substitute in the initial equation each of the solutions in the two members:

With x=3133,87:

they are equations with radicals
Both members have the same result. Then this solution is valid.

With x= 2948,73:

examples radical equations

Here, every member has a result, therefore, this solution is not valid.

The correct result is 10307,8.

How to solve equations with two radicals

So far we have seen how to solve equations with radicals, in which we had only one term with radicals.

But how do you solve equations with two radicals?

To solve the equations with two radicals, we must repeat the process 2 times, in order to eliminate the two radicals.

At the end, we must check which solution is valid, just like for equations with a radical.

Let’s see an example:

radical equation exercises

When we have two radicals in an equation, we must begin by leaving one of the races in only one of the members.

In this equation we already have one radical only, so we squared both members:

How to make equations with radicals

In the first member the root is annulled with the square and in the second, we develop the remarkable product, where one of the terms is the other root:

solve equations with roots

We operate on every term. When the root is squared in the remarkable product disappears:

equuacion radical

We are left with an equation where we only have one radical, so we have to repeat the process once again.

We leave the root alone in the second limb:

index 2 radical equations

We simplify terms:

equation with radicals

We squared both terms:

equation exercises with solved radicals

In the first term we develop the remarkable product. In the second term we squared both factors of the parenthesis:

equations with roots exercises

We eliminate the parenthesis of the second member by multiplying both terms by 16:

equations with solved radicals

We have a second-degree equation. To solve it, we pass all terms to one member and equal zero:

equations with radical examples

We apply the general formula for solving second-degree equations:

equations with two radicals

equations with square roots

And we get two solutions:

equations with solved roots

which is a radical equation

Let’s check which of the two is valid.

For x=61, we substitute in the original equation and operate:

equations with two radical examples

equations with two radical exercises solved

Equality is not met, so x=61 is not a valid solution.

For x=5:

exercises equations with two radicals

How to solve equations with two radicals

Equality is met, so x=5 is the solution to the equation.