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Quadratic Equations and Nature of Roots

In Secondary 3 E Maths, we learn that in order to solve a quadratic equation in the form \(ax^2+bx+c=0\), we can use the quadratic formula:

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\).

This formula has a few interesting symbols that affect the number of solutions, also called "roots", which we can get from a quadratic equation. In this article, we’ll go through how to find and describe the nature of the roots of a quadratic equation.

The most prominent symbol in the formula is the square root. We cannot obtain a real number by square-rooting a negative number. Thus, the key part of the formula that determines how many real roots we can get is the expression within the square root, \(b^2-4ac\). We call this expression the discriminant, \(D\), of the quadratic equation. There are three different cases for the three different natures of roots:

1\(b^2-4ac>0\) : 2 real and distinct real roots

A positive discriminant, \(D\), means that \(\sqrt{D}\) is a positive real number. 

The '\(\pm\)' sign in the formula means we either add or subtract \(\sqrt{D}\).

Thus, \(x = {-b + \sqrt{D} \over 2a}\) or \(x = {-b - \sqrt{D} \over 2a}\).

2\(b^2-4ac=0\) : 2 real and equal roots

When \(D=\) \(0\)\(x = {-b \pm \sqrt{0} \over 2a}\).

Thus, \(x = {-b \over 2a}\).

Take note that, in this case, the number of roots is still considered as two, both of which are equal.

3) \(b^2-4ac<0\) : No real roots

As mentioned, a negative discriminant, \(D\), means that \( \sqrt{D} \) is undefined. Technically speaking, it is still possible to square root a negative number to get “non-real” numbers which we call "complex" or "imaginary" roots. However, the current syllabus does not require us to learn how to obtain these roots. 

When we are describing the nature of roots, it is important to include the word “real” to emphasise that there are no real-number solutions to the equation.

There are many ways in which questions can test the topic of the nature of roots. Sometimes, a quadratic equation may be given and you are required to determine the nature of roots of the equation by evaluating the discriminant. Other times, you may be required to determine the value or range of values of an unknown given a quadratic equation and its nature of roots specified in the question.

Ultimately, remember that the discriminant is derived from the quadratic formula and it's useful in helping us determine the number of roots of a quadratic equation without actually solving it.

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