The quadratic equation is one of the important topics studied under algebra in the Senior High School years. Understanding the various methods of solving the quadratic equation is prerequisite to learning the many applications of this important algebraic equation.
The general quadratic equation is expressed as a x2 + b x + c = 0 , where a, b, c are constants ( with the qualification that a must be non-zero, otherwise this won’t be a quadratic equation at all ). In the equation, x is our usual unknown, whereas a, b, c are known as ‘coefficients’ in math language. The term ‘quadratic’ means that generally the equation has a second root, other than the one root for the more elementary linear equation such as a x = b, in which case the solution or a single root is x = b/a, so long as a is not zero.
DISTINGUISHING BETWEEN THE QUADRATIC EQUATION FROM A QUADRATIC EXPRESSION/FUNCTION.
The expression a x2 + b x + c is just a quadratic expression; if it satisfies the usual rules pertaining to an algebraic function, it could be a quadratic function and we could represent the function is the following way:-
f : x a x2 + b x + c or sometimes we write y = a x2 + b x + c to indicate that a x2 + b x + c is a quadratic function or expression and it can be represented graphically.
Those who have done some high school algebra should have had practices in graphing the quadratic function: by plotting values of x on the horizontal axis and values of y on the vertical axis, one would obtain a curve which is known as a ‘parabola’ with a vertex, either upright or inverted and with or without part of the parabola cutting the x-axis.
Of course, the quadratic equation a x2 + b x + c = 0 is related to the quadratic function – for instance, the roots of the quadratic equation can be found by plotting the graph of the relevant quadratic function and determining where the parabola cuts the x-axis. This is one way to solve or find the roots of the quadratic equation.
e.g 1. To solve the equation x2 + x – 2 = 0, we could graph the function y = x2 + x – 2 and from the graph obtained, we could determine the roots of this quadratic equation by noting that the parabola cuts the x-axis at x = 1 and x = -2 and therefore the roots are x = 1, -2.
For the discerning reader, you would have noticed that a quadratic equation always yields 2 roots. But this is true only if the equation is solvable at all. We will find out later that not all quadratic equations are solvable – in the sense that not all quadratic equations will yield ‘real’ roots as some roots are ‘imaginary’ or ‘complex’ mathematically speaking. And incidentally, for such cases if a graph of the relevant quadratic function is plotted, the parabola obtained will not intercept or cut the x-axis at all!
THE ALGEBRAIC METHOD OF SOLVING A GENERAL QUADRATIC EQUATION.
We have just learned the graphical method of solving quadratic equations; it is however, not a very convenient way of solving or finding the roots. Mathematicians ( from early European, to Persian, Arabic as well as from the Far East – China, India, etc ) had independently ‘discovered’ an algebraic method of solving the equation. The formula arrived at could be memorized but it is better to show how this formula is arrived:-
We begin with the usual quadratic equation a x2 + b x + c = 0
Since a is not zero, we divide throughout by a, and obtain x2 + (b/a)x + c/a = 0
Moving the constant term c/a to the other side of the equation, we get x2 + (b/a)x = – c/a
We now add the term (b/2a)2 or b2 /4a2 to both sides of the equation, bearing in mind that in an equation, whatever we do on the left hand side (LHS) we must do the same on the right hand side (RHS).
We thus obtain x2 + (b/a) x + b2 /4a2 = – c/a + b2 /4a2 . We now note that the LHS of this equation is a ‘perfect square’ in that the LHS expression could be rewritten as ( x + b/2a)2 and the RHS could also be rearranged; we thus obtain the following
( x + b/2a)2 = (b2 – 4ac)/4a2
Following the rule on finding roots of simple quadratic equations such as x2 = 9 would yield x = +3 and -3
We would get ( x + b/2a ) = +sq root of (b2 – 4ac) /2a and – sq root (b2 – 4ac) /2a
Bringing the term b/2a from the LHS to the RHS, we thus obtain the formula for the roots of a general quadratic equation as
x = [ -b + or -sq root of (b2 – 4ac) ]/2a …..derived formula
Of course one could memorize the formula but I would strongly advise students to learn how it has been derived, as shown above.
THE IMPORTANCE OF THE TERM b2 – 4ac .
In algebra, the expression b2 – 4ac is known as the ‘discriminant’ and its value is of great importance as it determines the nature of the roots to a quadratic equation. We shall discuss the following possibilities and the attendant consequences viz:
When the discriminant b2 – 4ac is positive.
In such a case, the roots of the quadratic equation are real and distinct, or real and different.
e.g 2 For the quadratic equation 2 x2 + 5x + 2 = 0, the discriminant is, with values of a = 2, b = 5 and c = 2, using the expression b2 – 4ac, we obtain 52 – 4(2)(2) = 25 – 16 = 9, a positive number.
This means that the equation has two roots which are real ( or non-imaginary, non-complex ) and distinctly different. In fact, using the formula or by factorization [ the expression 2 x2 + 5x + 2 may be factorized as ( 2x + 1 )( x + 2) ], the roots turn out to be x = -2, -1/2
It is also instructive to note that the graph of the function y = 2 x2 + 5x + 2 will be a parabola that intercepts the x-axis at the points x = -2, -1/2 , corresponding to the roots, of course.
When the discriminant b2 – 4ac equals zero.
In such a situation, the roots of the relevant quadratic equation will be real and identical, or we can also say that the roots are ‘repeated’.
e.g. 3 In the quadratic equation 9 x2 – 12 x + 4 = 0, with a = 9, b = – 12 and c = 4, the value of the discriminant would be b2 – 4ac = (-12)2 – 4 (9)(4) = 144 – 144 = 0. By factorization, or by using the derived formula above, the roots are x = 2/3, 2/3 i.e. two real roots, or one real root which has been repeated.
A graph of the quadratic function y = 9 x2 – 12 x + 4 will yield a parabola which ‘touches’ the x-axis at the point x = 2/3; in math language, we say the x-axis is a tangent to the parabola.
When the discriminant b2 – 4ac turns out to have a negative value.
When this situation obtains, the roots of the relevant quadratic equation do not exist in the real field; instead the roots are ‘imaginary’ or ‘complex’.
e.g. 4 The quadratic equation x2 – 2x + 5 = 0, with a = 1, b = -2 and c = 5, will yield a value of (-2)2 – 4 (1)(5) = 4 – 20 = – 16, which is a negative number. It is interesting to evaluate the roots using the derived formula above.
We would obtain the value x = [ -b + or -sq root of (b2 – 4ac) ]/2a = [ -(-2) + or -sq root of (4-20)]/2
And we finally obtain the value
x = [2 + or – sq root of -16]/2
Now what is the significance of the square root of a negative number such as negative 16? Mathematicians have long envisaged such a situation and have called the root of – 1 or -1 the imaginary number or i.
So for readers who have been exposed to complex and imaginary numbers, the roots of the equation in this example may be stated as x = 1 + or -2 i, where i is -1.
For completeness it is noteworthy that the graph of the function of y = x2 – 2x + 5 will show a parabola that will not intercept the x-axis at all! This is the result because, in tandem with the fact that this equation has no real roots – its roots are complex – the curve or the parabola representing the function must be cut the x-axis, for if it does, then there must be real roots.
The quadratic equation is one of the most essential parts of a course in high school algebra and an understanding of this equation and the associated quadratic function is vital: in many fields such as engineering and physics, as well as other physical sciences, there are quite a lot of situations in which solving quadratic equations becomes necessary. The understanding will help the student appreciate and enable him or her to learn to apply the techniques of solving quadratic equations in real-life situations.