How to solve second-degree equations step by step. Resolved exercises.

Below we will explain the basic steps to help you learn how to solve all the second degree equations, both complete and incomplete. Second-degree equations are also known as quadratic equations.

Complete second-degree equations

In general, second-degree equations are those where the x appears elevated to 2 in one of its terms.

They can be complete or incomplete second-degree equations, depending on whether they all have their terms or not. Here I’m going to focus on explaining the full second degree equations.

What are complete second-degree equations

The complete second degree equations or quadratic equations are represented as follows:

ecuación de segundo grado completa

Where a, b and c are the constants of the equation:

  • a is the number that always goes in front of x squared
  • b is the number that always goes in front of the x
  • c is the number without unknow

That is, the complete second degree equations are those that have an endpoint with x elevated to 2, term with x elevated to 1 (or simply x). If any of these terms are missing, we would be talking about incomplete second-degree equations, which are solved by a different procedure.

As second-degree equations, they have 2 solutions. Remember that the degree of an equation is equal to the number of solutions.

How to solve complete second degree equations

Identification of constants in the second degree equation

The first step in solving complete second degree equations is to identify the constants correctly. As we have said before, constants are the numbers that go in front of x squared, x and the term that does not carry x.

Let’s take an example:

resolver ecuaciones de segundo grado

In this case, in front of x squared, there is nothing, therefore a = 1.

In front of x there is a 5, so b=5.

And the term that does not carry x is a 4, so c=4.

Remember that when there is nothing in front of the unknowns, it is because they are multiplied by 1, or in other words, it means that there is a 1 in front.

Let’s look at another example:


fórmula para resolver ecuaciones de segundo grado

Now, if we notice, the equation is slightly different, but it is cause of many errors if we are not careful. Let’s see why:

In the general form there are no less signs:

ecuación de segundo grado

Therefore, we must transform our equation so that it is in the same way as the general form of the complete second degree equations:

formula general raíz negativa

Now we have it in the same way, where no minus sign appears and then a, b and c is obtained as in the first case:

ecuación de segundo grado fórmula

When we have more practice, we will identify constants directly, without needing to transform our equation, but to begin with it is a very good way to avoid mistakes.

General formula of complete second degree equations

Once the constants have been identified, the following formula must be applied to solve the complete second degree equations:

como hacer ecuaciones de segundo grado

Let’s see how it is used, solving the previous examples.

We have the first equation of second degree, in which we have identified the constants:

cuales son los metodos para resolver ecuaciones de segundo grado

Now, we have to replace the value of each record in the general formula:

como hacer una ecuación de segundo grado

And now we operate, within the root, taking into account the hierarchy of operations:

como resolver ecuaciones cuadraticas completas

At this point, we have to resolve the + sign on the one hand and the – sign on the other:

pasos para resolver ecuaciones cuadráticas

Then the two solutions would be -1 and -4. If we had the case that the fractions were not exact, they would have to be simplified.

Be very careful with the signs less than the constants.

There are particular cases where the result of the root is negative, or that its solutions are not exact or the result of the root is not exact.

Second-degree equations solution

Second-degree equations with root solutions

Solutions to a second-degree equation do not have to be two different integers. In some cases, they may have a double or two complex solutions.

Many times when the solutions are not complete, you begin to doubt if your solution is correct or not.

And now, let’s see how the solutions of a second-degree equation can be.

We come across this case when the root does not have an entire solution. As a general rule, it will be left in the form of a root so as not to have to operate with decimals, although if we are solving a problem and the exact result is needed, we will have no choice but to solve the square root with the calculator.

For example, we have the following second-degree equation:

ecuaciones de segundo grado ejercicios resueltos formula general

The first step is to solve the second degree equation using the general formula:

ecuaciones de segundo grado por formula general

At this point, we see that the root of 5 has no exact solution. Therefore, mathematically, it is left in the form of a root:

ecuaciones de segundo grado fórmula general

It is not mandatory to leave it in root form, but it is more comfortable to leave it like this, so you don’t have to drag decimals. The result could be given with decimals and would be just as correct. It is the same as with fractions, which when a result is not accurate, it is left in the form of a fraction.

Second-degree equations with double solution

When we are solving a second-degree equation and the root or discriminant result is 0, it is said that we have a double solution, since we are going to have the same solution repeated twice. Let’s see how to act in this case:

ecuaciones cuadráticas con soluciones complejas

As a priori we do not know how the solutions are going to be, we solve the equation like any other:
cuando una ecuación de segundo grado no tiene soluciónWhen we get to this part of the resolution, we see the root result is 0.

A serious mistake is to leave only 1 solution. Never do this because you will be suspended directly. What you do in these cases is work with 0:

reglas para resolver ecuaciones de segundo grado

It is resolved following the usual procedure, although it seems obvious to add and subtract 0, but it is a good way to reach the 2 solutions.

Another way of indicating the solutions is to develop the general formula to the end, to arrive at a solution but to indicate in writing that it is a twofold solution.

Second-degree equations with complex solutions

We come across this case when in the general formula, the discriminant or root result is negative.

If you haven’t studied the complex numbers yet, when you get to negative root you should put the following:

It has no real solution

This sentence is equivalent to saying that within the set of real numbers, there is no solution (the solution is found in the set of complex numbers).

Let’s look at it with an example:

raíz cuadrada negativa en ecuación de segundo grado

ecuaciones cuadráticas

In other words, as soon as we see the root with a negative content, we directly indicate that there is no real solution and that’s it. It is important not to forget the real word, because if you simply indicate “no solution”, it will not be correct, because it does have a solution, but not in the set of real numbers.

On the other hand, if you have already studied complex numbers, you have to develop the equation until you find complex solutions. That is, you must replace the root of -1 with the number i:

ejemplo de ecuación de segundo grado

ejemplos de ecuaciones cuadráticas

Below we will explain the basic steps to help you learn how to solve incomplete second degree equations.

What are Incomplete Second Degree Equations

Let’s remember the general form of a complete second degree equation:

1 ECUACIONES DE SEGUNDO GRADO COMPLETAS

The incomplete second-degree equations are those in which the constants b or c or even both are missing, i. e. equal to 0. We got three guys:

When b=0:

ecuacion de segundo grado incompleta 2

When c=0:
ecuacion de segundo grado incompleta 1

When b = 0 and c = 0:
ecuacion de segundo grado incompleta 3

Let’s see how each one of them is solved.

How to Solve Incomplete Second Degree Equations

Endless incomplete second-degree equations with x (b=0)

We are faced with an equation of this type when the second-degree equation lacks the term with x or in other words, when b=0:

ecuacion de segundo grado incompleta 2

For example:

ecuacion segundo grado incompleta b=0

To solve incomplete second-degree equations of this type, we first clear x² as if it were a first-degree equation:

resolución ec segundo grado incompleta

Once here, we have to move the square to the other side of equality as a root, and then get a positive and a negative solution:

soluciones ec segundo grado incompleta

Whose solutions are 2 and -2.

That’s it, just like that. If in another equation, the root does not give exact, each result is left as a root with its corresponding sign in front of it.

Incomplete second-degree equations without number (c=0)

The incomplete second-degree equations without number (or without an independent term) are those where c=0 in the general form and therefore have this form:

ecuacion de segundo grado incompleta 1

For example:

ecuaciones de segundo grado incompletas 5

The first step in solving this type of incomplete equations is to draw a common factor, since an x is repeated in both terms.

ecuaciones de segundo grado incompletas 6

Now, we must consider the following:

Cuando una multiplicación de dos factores tiene como resultado 0, quiere decir que uno de los 2 factores es 0, ya que cualquier valor multiplicado por 0 es 0.

 

For example:

2.0 = 0 (this is clear)

x. 0 = 0 (x can be any value, but when multiplied by 0, the result is 0)

a. b = 0 (the result here is 0, so either a=0 or b=0, but we don’t know which one)

We continue with our equation. We have a case similar to a. b = 0: we have two factors (x and (x-3) whose result is 0, so one of the two must be 0, but we don’t know which one.

Therefore, we have two paths to follow: x=0 or x-3=0. In the first case, we get directly the first solution and in the second case, we have to take another step, which is to clear the x:

ecuaciones de segundo grado incompletas 7

The solutions are x=0 and x=33

You have to be very careful with negative signs when drawing common ground.

Incomplete second-degree equations with x² only (b=0 and c=0)

This is the easiest type of all that is almost directly solved. Incomplete second-degree equations of this type are those that only have the term x², or in other words, when b=0 and c=0:

ecuacion de segundo grado incompleta 3

Let’s look at an example:

ecuaciones de segundo grado incompletas 8

As in the case of incomplete second-degree equations with c=0, we must clear x²:

ecuaciones de segundo grado incompletas 9

But, the peculiarity of this case is that we will always reach x²=0. So, when we move the square to the other side as root, we have that the result is 0, but it’s a double solution:

ecuaciones de segundo grado incompletas 10

 

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