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

Let us rewrite the quadratic equation as ( X-R1) * (X-R2) = 0. The above step is termed as factoring.

Let us also rewrite the original generalized equation of the quadratic as X*X + B/A * X + C/A = 0

The factored equation can be rewritten 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 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 which is equal to the 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. When plotted it will yield the equation of a parabola.