The Nature of Roots, or more comprehensively Nature of roots of quadratic equation, is an important concept that outline why the quadratic equation is important for different advanced mathematical discussions. The Number line, when discussed mathematically, discusses just the real number system in most cases.
Most high school students will only consider numbers from the eye of the real system. These numbers comprise Rational numbers, irrational numbers, natural numbers and the more encompassing number Integers that consider the negative parts of the natural number.
While these numbers above represent most of the digits that students will ever use in mathematics, imaginary numbers also play a major role. An imaginary number is a type of number that does not exist in the real number system. The imaginary number is born from a negative square root. For example, the value
is an imaginary number. In fact, the number is the basic imaginary unit on which other negative numbers in square roots are found. Now, if students are to input
Into a calculator, they will get a syntax error.
Now the imaginary number is not in itself a complete number. It will only be complete with the help of a real number to form the complex number.
Quadratic equation is the major source of the imaginary number. Certain quadratic equation solutions or roots will lead to either a real or imaginary number. Even if a real number is formed, the root will have to fall inside the rational, irrational, natural or integer category. The ability to identify which category the quadratic equation falls into is what is known as the Nature of Roots.
What Is the Nature of Roots
The nature of roots is the category of any real or complex numbers under which the roots or solutions of a quadratic equation fall. For example, if the solution of a quadratic equation ends up being 2 and 3, then it is a real number and also a rational number. While this example is simple, it is important for students to understand that they do not need to solve to the solution before they determine the nature of roots of their quadratic equation.
It is important to understand that an equation’s number of roots depends on the type of polynomial being handled. That is, if students are dealing with a polynomial of degree 3, they are naturally supposed to end up with 3 roots or solutions. Since a quadratic equation has a degree of 2, students will always have two roots.
If the two roots reached to have the same value, it means that they are equal. For the roots to be unequal, the values of x and y will not be the same.
While finding the nature of the roots calculator is possible, understanding the basics of the calculation and what it means will help you know what it stands for.
Nature of Roots: The Different categories of Roots
Knowing the different categories of roots will help students understand the nature of their roots once they find them.
Now, the quadratic equation is given as
ax2 + bx + c = 0
Now, finding the quadratic equation highly depends on the General quadratic formula, which is given as
The respective a, b, and c in the above formula are the same as the equation ax2 + bx + c = 0
The two solutions or roots that will be reached using the general formula will be. Now the two solutions are given as
Now the expression under the square root of the general formula, which is b2 – 4ac (shown in red) is what can help students find the nature of their roots. The expression is called the determinant of the quadratic equation as it determines the type and nature of roots that will be derived at the end. As already stated, students will not need to reach a final calculation before they know the nature of their roots.
So here is how to determine the nature of roots
Now, a, b and c are naturally real numbers when it comes to quadratic equations where a ≠ 0. We will assume that these two conditions are consistent
When Nature of Roots Are Real and Unequal
If b2 – 4ac > 0, it means that the discriminant is positive, and the square root under which it is based in the quadratic formula will also turn out positive. It also means that the roots x and y which will be derived from the quadratic equation will be real and unequal.
As an example, it means if for any time x = 5, y ≠ 5
- When Nature of Roots Are Real and Equal
If b2 – 4ac = 0, it means that the discriminant is zero, and the square root under which it is based in the quadratic formula will equal zero. It also means that the roots x and y which will be derived from the quadratic equation, will be real and equal.
As an example, it means if for anytime x = 5, y = 5
- When Nature of Roots Are imaginary and Unequal
If b2 – 4ac < 0, it means that the discriminant is lesser than zero, and the square root under which it is based in the quadratic formula will not be identifiable. This is because the negative of a square root cannot be found. It means that the roots x and y which will be derived from the quadratic equation, will be imaginary and unequal. These particular roots will invite the complex lane and complex numbers into the equation.
It is important to understand that a number in any of the forms is not a real number because there is n square root of a negative number mathematically. The only way they can be solved is through the use of the complex number theory.
- When Nature of Roots Are Real, Rational and Unequal
If b2 – 4ac > 0 and forms a perfect square, it means that the discriminant is positive and will give a perfect square like 4, 9, 81 and so on. This by extension, means that the square root of the quadratic formula will lead to a square root that is a complete number without fractions or decimals. For example, the square root of a perfect square 9 will give a whole number 3 without any decimals.
A positive discriminant that is a perfect square means that the roots x and y which will be derived from the quadratic equation, will be real, rational and unequal.
- When Nature of Roots Are Real, Irrational and Unequal
If b2 – 4ac > 0 and does not form a perfect square, it means that the discriminant is positive and will not give a perfect square like 4, 9, 81 and so on. This, by extension, means that the square root of the quadratic formula will not give a square root that is a whole complete number. The roots will either be in decimals. An irrational number generally means numbers in decimals.
- When Nature of Roots Are Simply Irrational
If b2 – 4ac > 0, forms a perfect square but a and b are irrational, then the roots that will be derived from the quadratic equation will be irrational.
This article was designed to help students understand exactly how the roots of quadratic equations work so that they can easily understand whatever numbers they arrive at after computations. While there are not many talks about the nature of roots.
Knowing that it is possible to arrive at any number when solving the quadratic equation will leave students more willing to explore any computations. Also, since the quadratic equation is highly connected to the complex number, it is very necessary for students to know how to change a negative value in a square root to an imaginary number.
This piece comprehensively stated the possible answers that students can get as solutions or roots when calculating Quadratic equations. All they need do is practice more to let the information outlined in this work stick.