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.
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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:
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:
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:
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:
The new equation looks like this:
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:
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
Y of each value of t, 2 values of x will result, having the 4 final values of the equation:
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:
1 – We change the variable:
We are left with the following complete second degree equation, with the t:
2 – We solve it by applying the general formula for solving complete second degree equations:
We separate as always the positive sign from the positive sign to calculate the two solutions of t:
3 – We undo the variable change by clearing the x:
As we have two values of t, we have to apply the formula for each value of t, from which we get 4 solutions:
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: