How to Factorize a Quadratic Equation in the Form x^2 - x - m1m2

When faced with the task of factorizing a quadratic equation in the form (x^2 - x - m1m2), one approach is to find numbers (a) and (b) such that their product equals (-m1m2) and their difference is equal to (-1). Let's delve into the detailed steps and reasoning behind this approach.

1. Understanding the Quadratic Expression

The quadratic expression we aim to factorize is in the form (x^2 - x - m1m2). To solve such an equation, we need to find two numbers (a) and (b), where:

(a) and (b) are such that (ab -1) (a) and (b) are also such that (ab m1m2)

This might initially seem contradictory, but let’s proceed by ensuring that the values of (a) and (b) fit both conditions.

2. Finding the Numbers (a) and (b)

The key is to find two numbers whose product is (-m1m2) and whose difference is (-1). Let's denote these numbers as (m2) and (m1), where (m2 - m1 -1).

Given this, we can start by recognizing that if (m2 - m1 1), then (m2 m1 1). This helps us in identifying the values of (a) and (b).

3. Expanding the Quadratic Expression

To factorize (x^2 - x - m1m2), let’s assume it can be expressed in the form ((x - a)(x b)). Expanding this form, we get:

[x^2 - (a - b)x - ab]

Comparing this to the original equation (x^2 - x - m1m2), we need to match the coefficients:

(-(a - b) -1) (-ab -m1m2)

From the first condition, we get:

[a - b 1]

And from the second condition, we get:

[ab m1m2]

With these two equations, we are closer to finding the values of (a) and (b).

4. Solving the Equations

Given (a - b 1) and (ab m1m2), we can substitute (b a - 1) into the second equation:

[a(a - 1) m1m2]

This simplifies to:

[a^2 - a - m1m2 0]

Solving this quadratic equation for (a) using the quadratic formula:

[a frac{1 pm sqrt{1 4m1m2}}{2}]

Once we have the value of (a), we can find (b) using (b a - 1).

5. Conclusion and Solution

The solution for the factorization of (x^2 - x - m1m2) is given by substituting the values of (a) and (b) found above:

[(x - a)(x b) left(x - frac{1 sqrt{1 4m1m2}}{2}right)left(x - frac{1 - sqrt{1 4m1m2}}{2}right)]

This form represents the fully factorized quadratic expression.

In summary, the key steps to factorize a quadratic equation in the form (x^2 - x - m1m2) are to identify the numbers (a) and (b) such that their product is (-m1m2) and their difference is (-1). By solving the equations derived from these conditions, we can find the factorized form of the quadratic expression.

Keywords

quadratic factorization, difference of products, algebraic equations