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.