The Theory of Quadratic Equations

The Theory of Quadratic Equations
By Srinivasa Gopal

A quadratic equation is a polynomial equation of second order. A quadratic equation has two roots. The roots can also be equal and identical. Let us write the quadratic equation in two forms

AX * X + BX + C = 0 an example of a quadratic equation would be 5X*X + 3 *X + 2 = 0

Rewrite this as ( X-R1) * (X-R2) = /0. The above step is termed as factoring.

The polynomial form of the equation is X*X + B/A * X + C/A = 0

The factored equation can be split as X * X -X( R1 + R2) + R1R2 = 0.

Comparing Similar terms we can see that -(R1 + R2) = B/A

R1R2 = C/A

(R1 + R2) = -B/A

Let us investigate the discriminant b *b - 4 * a * c

b = -a ( r1 + r2)

c = AR1R2; 4*A*C = 4 * A* A* R1 * R2

b*b = A*A(R1 + R2) * (R1 + R2)

DISCRIMINANT = A*A(R1 + R2) * (R1 + R2) - 4*A*A*R1*R2

= A*A ( (R1+R2)((R1+R2) - 4R1R2)

= A*A ( R1 - R2) * (R1 - R2).

Notice that this is a perfect square of A(R1-R2). So if the discrimant becomes negative it means that the quadratic equation does not have real roots as squares of real numbers are also perfect squares.

Let us add A( R1-R2) to -b which is A( R1 + R2), and the sum is 2AR1. Dividing this by 2A would yield R1.

Similarly let us subtract A( R1-R2) from -b ie., A( R1 + R2) - A (R1-R2)

which is equal to A(2R2) or 2AR2. Dividing this by 2A would yield R2.

So R1 is (-B + squareroot( discriminant) ) / 2A and R2 is (-B - squareoot( discriminant) / 2A

Let us take some common factoring problems that you would encounter

say x * x + 5*x + 6 = 0.

First step evaluate the discriminant = SQUAREROOT(25 - 24) = 1, which means that there are real roots.

The roots of the equation are (- 5 + 1)/ 2 is equal to -2 and ( -5 -1)/2 equal to -3.

The equation can be factored as (X+2)(X+3) = 0.

Let us take another example

3 * x * x + 9 * x + 6 = 0, rewriting this as x * x + 3*x + 2 = 0.

discriminant = sqrt(9-8) = 1
R1 = -1 and R2 is -2. So the factored form of the same equation is

(x + 1)( x+ 2) = 0.

A quadratic equation can also be plotted on a graph. It will yield the equation of a parabola.

The author is an Ezine Expert on issues related to science, IT and Cricket. Known as Srinivasa Gobal to my friends and family prior to 1992 and known as Srinivasa Gopal and Gopal Srinivasan later during 1999- 2008.

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