The quadratic equation
Quadratic equations are the type of equations shown in the form below:
where are real numbers. Every (real or complex) that satisfies this equation is called the solution of the quadratic equation.
If , our equation will be equal to which transforms into a linear equation. This means that must be different from zero.
Number in this equation is called the leading coefficient, number is the linear coefficient and free coefficient.
There are three special forms of quadratic equations:
1.) If and then the equation is:
First, from the left and right sides of an equal sign we subtract and then divide the entire equation by . We now get an equivalent equation:
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 solutions:
a) If , then the solutions are real numbers and .
b) If , then the solutions are complex numbers and .
Example 1. Solve the equation .
and we got . As we know , but also which means that our solutions are and .
Example 2. Solve the equation .
The procedure is the same, and quickly enough we arrived at the point where . This means that solutions of our quadratic equation are not real numbers, and are only complex: , .
2.) and .
The quadratic equation is now given in the form:
This is one of the simplest forms of a quadratic equation because it can be written as a product of two linear equations, where one solution is always equal to and other is equal to .
Example 3. Solve the equation .
First, we will extract . Then we have . Now we have a product of two numbers that is equal to . This means that one of them has to be zero. The first case is where equals to zero, and second is where equals to zero. This directly leads us to our two solutions which are and => => .
3.) and .
In this case the quadratic equation looks like:
This equation is valid iff is . We will say that this quadratic equation also has two solutions which are equal, that is .
is called a double solution of the quadratic equation .
The solutions of the quadratic equation
We observe 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 4. Solve the equation .
First we determine our coefficients: , and . Now we simply inserted it into the formula for the solutions of the quadratic equation:
If we insert the solutions we obtained into the given equation, left side of the equation must be equal to .
This is the universal method of finding solutions, however, in certain cases we can make it a lot easier. One way of simplifying is factoring. If we have two factors whose product is equal to zero we can easily find solutions just like in the third special form of the quadratic equation we observed.
Solve the equation by factoring.
So, is the double solution of an equation .
Solving quadratic equations by completing the square
Recall the formula for the square of a binomial:
to proceed further.
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 .
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:
How can this help us solve quadratic equations?
Example 8. Solve the equation by completing the square.
From the previous example above it was expressed that the quadratic equation can be written as:
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.
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:
If is one solution to the equation , what is the other solution and what is ?
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 .
Determine the product and sum of the solutions of equation .
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 .
Solve the following system of linear and quadratic equation:
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 :
The solutions of the quadratic equation are:
Now we must solve linear equations and .
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:
The quadratic equations worksheets
Solve by taking square roots (222.8 KiB, 233 hits)
Solve by factoring (466.1 KiB, 309 hits)
Completing the square (345.0 KiB, 282 hits)
Solve using quadratic formula (308.2 KiB, 265 hits)
Find a discriminant (473.2 KiB, 182 hits)
Factoring quadratic expressions (315.0 KiB, 206 hits)