The Theory of Quadratic Equations

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.

Source by Srinivasa Gopal

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