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- The Theory of Quadratic Equations
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## The Theory of Quadratic Equations

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

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

We rewrite the quadratic equation as: ( X-R1) * (X-R2) = 0. The above step is called factoring.

We also rewrite the original generalized equation of the root mean square as X*X + B/A * X + C/A = 0

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

Comparing like terms, we see that -(R1 + R2) = B/A

R1R2 = C/A

(R1 + R2) = -B/A

Let’s study B* B – 4 * A * C

B = -A (r1 + r2)

C = AR 1 R 2 ; 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).

Note that this is a perfect square of A(R1-R2). Therefore, when the discriminant becomes negative, it means that the quadratic equation has no real roots, since the squares of real numbers are also perfect squares.

We add A( R1-R2) to -B, which is A( R1 + R2), and the sum is 2AR1. Dividing this by 2A gives R1.

Similarly, we subtract A( R1-R2) from -B, i.e. A( R1 + R2) – A (R1-R2)

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

So R1 is (-B + square root( discriminant) ) / 2A and R2 is (-B – square root( discriminant) / 2A

Let’s take a look at some common factoring problems you’ll encounter

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

As a first step, evaluate the discriminant equal to SQUARE ROOT(25 – 24) = 1, which means there are real roots.

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

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

Let’s take another example

3 * x * x + 9 * x + 6 = 0, rewriting it 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 graphed. When drawn, the equation of the parabola is obtained.

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