How to solve two-square equations step by step. Solved exercises.

In this section we will explain what two-square equations are and how to solve two-square equation exercises step by step. We will also discuss other types of equations that are solved with the same procedure.

What are bi-square equations other than grade 4 equations?

The two-square equations are equations that have a form similar to the complete second degree equations.

We recall that the complete second degree equations have this form:

Two-square equations

The two-square equations differ from the previous ones in that the exponents are multiplied by 2, hence their name is two-square equations (twice a quadratic equation). This is its general form:

ecuación bicuadrada

These are fourth-degree equations, which have terms with x elevated to 4, x elevated to 2, and only with number. Being fourth-degree, it has 4 solutions.

Let’s see in the next section how to solve the two-square equations

How to solve two-square equations

The two-square equations are solved almost the same as the complete second-degree equations, but with the difference that it is necessary to make a change of variable first and undo this change at the end to obtain the four final solutions. Let’s look at it more slowly:

We start from the general form of a two-square equation:

equations elevated to 4

1 – We perform the variable change: This is done because we need the two-square equation to have the same shape as a complete second degree equation. The change of variable to be made is the following:

how to make two-square equationsThe new equation looks like this:formula two-square equations

2 – We apply the general formula to solve complete second degree equations to the equation with the new variable, from which we obtain two solutions:
equation elevated to 4

3 – We have obtained 2 solutions, but with our new variable t. We need to undo the variable change to get to the 4 x solutions of the original equation.

That is, from the change of x² = t, as we already know t, we clear the x

bicuadratic equations

Y of each value of t, 2 values of x will result, having the 4 final values of the equation:

solving two-square equations

equations with variable change

Let’s see an example now and solve it step by step

Solved exercises of two-square equations step by step

We start from the following equation:bi-square equations example

1 – We change the variable:

how to form a two-square equation

We are left with the following complete second degree equation, with the t:


bi-square equations explanation

2 – We solve it by applying the general formula for solving complete second degree equations:

How to solve fourth-degree equations

We separate as always the positive sign from the positive sign to calculate the two solutions of t:

equuaciones bicuadradas ejercicios

3 – We undo the variable change by clearing the x:

ejercicios equuaciones bicuadradas

As we have two values of t, we have to apply the formula for each value of t, from which we get 4 solutions:exercises solved two-square equations

Therefore the solutions of the two-square equation are 5, -5, 3 and -3

Other equations with variable change

The same general procedure can be used to solve equations that have this form, with the corresponding variable change:

solve grade 4 equations