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Quadratic equations are the type of equations in the form

where are  and .

If , our equation reduces to which is linear equation. More about solving linear equations you can find in lessons One-step equations, Two-step equations and Multi-step equations.

Number in this equation is called the leading coefficient, number is the linear coefficient and is constant.

Every (real or complex) that satisfies this equation is called the solution of the quadratic equation.

The formula for the solutions of the quadratic equation

The universal method of solving these kinds of quadratic equations is by using the formula for the solutions of the quadratic equation:

Example Solve the equation .

First we determine coefficients: , and . Now we simply insert coefficients into the formula for the solutions of the quadratic equation:

We can check if result is correct by inserting the solutions we obtained into the given equation:

and

If quadratic equation is in special form sometimes it is easier to manipulate given equation to find solutions instead of using formula for solutions of quadratic equations. Some of these special forms are:

1)  If  and then the equation is:

First, from both sides of equation we substract

and then divide the entire equation by

This equation always has two different solutions. The solutions may be real or complex numbers, depending on the sign of numbers and .
If then the quadratic equation has two real solutions:

If , then the solutions are complex numbers

Example Solve the equation .

We add to both sides of equations and obtain . Now we divide equation with 2 to get . Because and solutions are and , that is

Example Solve the equation .

The procedure is similar to the one in example above, we substract 8 from both sides and divide equation by 2 to obtain . This means that  solutions of this quadratic equation are not real numbers, they are complex: , .

2) and .

The quadratic equation is now given in the form:

We can extract from both terms and write equation as . Now notice that if or left side of equation is equal to zero. So is one solution and other solution is solution of linear equation . Solution of that linear equation is (we substracted from both sides and divided it by ).

Example Solve the equation .

First, we extract . Then we have  . One solution is . We find second solution by solving :

3) If and equaion reduces to and we immediately see that only solution is .

is called a double solution of the quadratic equation .

Factoring

Other way of simplifying is factoring. If we have two factors whose product is equal to zero we can easily find solutions. Suppose we have written quadratic equation as for some real numbers and . Then obviously and are solutions of original equation (if we insert or in given equation we see that equality holds).

Example Solve the equation by factoring.

Using the formula for square of sum (see lesson Determining polynomials, basic math operations, the most important rules for multiplying, under section Multiplication) we write as which can be written as so given equation is equivalent to . We now see that is double solution.

Solving quadratic equations  by completing the square

Useful technique for solving quadratic equations is completeng the square. We are going to tranform our original equation to more appropriate form from which we can easier find solutions.

First, we extract leading coefficient :

Now we add and substract :

and group terms in a way that allows us to use square of sum formula

So we have .

Algorithm above, with geometric interpretation is shown in animation below.

To find solution of we can find solutions of . We can substitute and and equation transforms to which we know how to solve from the first case of special form equations.

To understand better the process we just decribed look at the next example.

Example 6.

If we have a term such as we can express it as .

We have a first member squared, second doubled and third squared.

We can write that down as .

Example 7.

We are observing the quadratic equation .

Let’s write the equation now in form:

.

From the left and right side of the equation we previously subtracted . We apply a methodology of completing the square on the left side of the equation.

is not a whole square, but is. This means that we can first extract and get:

.

Now we have the first member squared. It can be observed that the square of the second member is missing, which is , because the first member when multiplied by is equal to . Also, we need to multiply by because the first member was multiplied by .

Therefore, now we have:

In the next step, the square from the second member, which is equal to , we added to both sides of the equation also multiplied by two:

How can this help us solve quadratic equations?

Example 8. Solve the equation by completing the square.

Solution:

From the previous example above it was expressed that the quadratic equation  can be written as:

Now we can calculate the square root of the whole equation:

.

## The discriminant of the quadratic equation

The discriminant of the quadratic equation is the number :

Then the solutions of the quadratic equation written by using the discriminant are:

If , then the quadratic equation thus has two distinct real solutions, if , the quadratic equation has two solutions that are complex conjugates , and if the quadratic equation has one real solution of multiplicity two.

## Vieta’s formulas

French mathematician François Viète also studied quadratic equation and came to an important relation between quadratic equation and system of two equations with two unknowns. Those unknowns are the solutions to observed quadratic equation.

The solutions  and  of the quadratic equation meet the criteria of the Vieta’s formulas:

Example 9.

If is one solution to the equation , what is the other solution and what is ?

Solution:

According to the Vieta’s formulas we find that:

Since we already have one solution this can be written as:

From the second equation we find that , which leads us to .

Example 10.

Determine the product and sum of the solutions of equation .

Solution:

From the equation we read coefficients:

.

Now, by using Vieta’s formulas we have:

.

## The system of linear and quadratic equations

The quadratic equation with two unknowns has the form:

where are real numbers – coefficients.

This kind of equation cannot be solved without any terms attached to it, however, if we have one more additional condition like a linear equation, it can be solved. This is because from the linear equation we get the information in which relation are unknowns and and we can then extract one using the other to get the quadratic equation with only one unknown. The solutions to this system are two ordered number pairs and .

Example 11.

Solve the following system of linear and quadratic equation:

Solution:

First we will observe a linear equation and extract one unknown according to the other one. It doesn’t matter which unknown we choose. In this case, we choose :

Now we take and insert it into the quadratic equation to get quadratic equation with only one unknown:

.

The solutions of the quadratic equation are:

.

Now we must solve linear equations  and  .

As follows:

The solutions are: , and .

Biquadratic equations are equations with two unknowns to the power of two. They are solved by using substitution.

Example 12. Solve the following system:

First we would like to have only one unknown in our equation so we will extract by and insert it in other equation:

Now we can make a substitution. We will remove of to the power of through substitution with to get the quadratic equation:

.

We are still not finished, we have to return to our substitution and solve the quadratic equation:

.

For we have:

.

For we have:

.

Solve by taking square roots (222.8 KiB, 421 hits)

Solve by factoring (466.1 KiB, 561 hits)

Completing the square (345.0 KiB, 539 hits)

Solve using quadratic formula (308.2 KiB, 462 hits)

Find a discriminant (473.2 KiB, 380 hits)

Factoring quadratic expressions (315.0 KiB, 391 hits)

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