**Factoring Quadratic Equations with a ≠ 1**

Quadratic equations in the form of ax^2 + bx + c = 0, where a, b, and c are constants, can be factored into the product of two binomials. However, when a ≠ 1, factoring becomes more challenging. In this article, we will learn how to factor quadratic equations with a ≠ 1.

**Why a ≠ 1 Matters**

When a = 1, factoring a quadratic equation is relatively straightforward. For example, consider the equation x^2 + 5x + 6 = 0. Using the factoring method, we can write:

x^2 + 5x + 6 = (x + 3)(x + 2) = 0

However, when a ≠ 1, the factoring process becomes more complicated. For instance, consider the equation 2x^2 + 5x + 3 = 0. In this case, we cannot simply split the middle term into two binomials.

**The General Method**

To factor a quadratic equation with a ≠ 1, we need to follow a general method that involves finding two numbers whose product is ac (the product of the leading coefficient and the constant term) and whose sum is b (the coefficient of the linear term). These numbers are often referred to as "factors" of ac.

**Step-by-Step Procedure**

Here's a step-by-step procedure to factor a quadratic equation with a ≠ 1:

### 1. Write down the quadratic equation in the standard form:

ax^2 + bx + c = 0

### 2. Find the product of a and c:

ac

### 3. Find two numbers, p and q, whose product is ac and whose sum is b:

pq = ac p + q = b

### 4. Rewrite the middle term bx as px + qx:

ax^2 + px + qx + c = 0

### 5. Factor the equation by grouping:

ax^2 + px + qx + c = ( ax^2 + px ) + ( qx + c ) = ax(x + p/a) + q(x + c/q)

### 6. Simplify the expressions inside the parentheses:

ax(x + p/a) + q(x + c/q) = (x + p/a)(x + q/a) = 0

**Examples**

### Example 1:

Factor the quadratic equation 2x^2 + 5x + 3 = 0.

**Solution:**

Following the step-by-step procedure:

### 1. Write down the quadratic equation in the standard form:

2x^2 + 5x + 3 = 0

### 2. Find the product of a and c:

ac = 2 × 3 = 6

### 3. Find two numbers, p and q, whose product is ac and whose sum is b:

pq = 6 p + q = 5

p = 2, q = 3

### 4. Rewrite the middle term bx as px + qx:

2x^2 + 2x + 3x + 3 = 0

### 5. Factor the equation by grouping:

2x^2 + 2x + 3x + 3 = (2x^2 + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1)

### 6. Simplify the expressions inside the parentheses:

2x(x + 1) + 3(x + 1) = (x + 1)(2x + 3) = 0

Therefore, the factored form of the quadratic equation is (x + 1)(2x + 3) = 0.

**Conclusion**

Factoring quadratic equations with a ≠ 1 requires a different approach than factoring those with a = 1. By following the general method outlined above, you can factor quadratic equations with a ≠ 1. Remember to find the two numbers whose product is ac and whose sum is b, and then apply the step-by-step procedure to obtain the factored form of the equation.