## A Journey into Quadratics: Solving through Factoring by Grouping

Dearest scholars, as we delve into the world of mathematics, we’re presented with the intricate dance of numbers and variables in quadratic equations. Today, we shall harness the art of factoring by grouping to unlock the their secrets.

### Factoring by Grouping: The Key to Quadratics

Often, quadratic expressions are not immediately factorable into binomial pairs, leaving one at an impasse. However, with the method of grouping, we can elegantly decompose and restructure these expressions, paving the way to their factorization.

In its essence, factoring by grouping involves splitting the middle term in a way that allows us to group the terms into pairs, and subsequently factor out a common element from each pair. This can be represented with a general equation of the form $$ax^2 + bx + cx + d$$.

Consider the equation $$x^2 + 5x + 6$$. This equation can be expressed as $$x^2 + 2x + 3x + 6$$. Here, we’ve divided the middle term, 5x, into 2x and 3x. Grouping the terms, we have:

$(x^2 + 2x) + (3x + 6)$

From this, we can factor out the common elements:

$x(x + 2) + 3(x + 2)$

Now, it’s clear that $$x + 2$$ is a common factor, yielding our final factorized expression:

$(x + 3)(x + 2)$

### Solving Quadratic Equations through Factoring by Grouping

Having factorized the quadratic expression, we can now set it to zero and solve for $$x$$ to find our solutions. For the equation $$x^2 + 5x + 6 = 0$$, since we’ve factorized it as $$(x + 3)(x + 2) = 0$$, our solutions are $$x = -3$$ and $$x = -2$$.

### Exercises

1. Solve $$x^2 + 9x + 14 = 0$$
Solution

Splitting the middle term: $$x^2 + 2x + 7x + 14$$.

Grouping terms: $$(x^2 + 2x) + (7x + 14)$$.

Factoring: $$x(x + 2) + 7(x + 2)$$.

Final factorization: $$(x + 7)(x + 2) = 0$$.

Solutions: $$x = -7$$ and $$x = -2$$.

2. Solve $$x^2 + 7x + 10 = 0$$
Solution

Splitting the middle term: $$x^2 + 2x + 5x + 10$$.

Grouping terms: $$(x^2 + 2x) + (5x + 10)$$.

Factoring: $$x(x + 2) + 5(x + 2)$$.

Final factorization: $$(x + 5)(x + 2) = 0$$.

Solutions: $$x = -5$$ and $$x = -2$$.

3. Solve $$x^2 + 4x + 3 = 0$$
Solution

Splitting the middle term: $$x^2 + x + 3x + 3$$.

Grouping terms: $$(x^2 + x) + (3x + 3)$$.

Factoring: $$x(x + 1) + 3(x + 1)$$.

Final factorization: $$(x + 3)(x + 1) = 0$$.

Solutions: $$x = -3$$ and $$x = -1$$.