Quadratic Equations and Nature of Roots
In Secondary 3 E Maths, we learnt 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 we can get from a quadratic equation, which we call roots. 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. As we should have learnt by now, we can’t square root a negative number. Thus, the key part of the formula that determines how many real roots we can get is the expression inside the square root, \(b^2-4ac\). We call this expression the discriminant of the quadratic equation. There are three different cases which correspond to 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) \(b^2-4ac=0\) : 1 real and repeated real root OR 2 real and equal roots
Since \(D=\) \(0\), we are either adding or subtracting \(0\), which gives us the same answer.
Thus, \(x = {-b \over 2a}\).
Take note that in this case, the number of roots is still considered as two, just that the two roots are the same.
3) \(b^2-4ac<0\) : No real roots
As mentioned, a negative discriminant, \(D\), means that \( \sqrt{D} \) will result in a math error. Technically speaking it is still possible to square root a negative number to get a “non-real” number, so we actually still have two roots. That is why 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. As long as you can simplify a problem into finding solutions to a quadratic equation in the form \(ax^2+bx+c=0\), don’t forget that the discriminant \(b^2-4ac\) will be a handy tool to quickly determine the number of solutions to the problem you should expect.
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