**Factoring Quadratic Equations: x^2 - 5x - 24 = 0**

In this article, we will discuss how to factor a quadratic equation, specifically the equation x^2 - 5x - 24 = 0. We will explore the concept of a perfect square trinomial and how to identify and factor it.

**What is a Quadratic Equation?**

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (x) is two. It has the general form:

ax^2 + bx + c = 0

where a, b, and c are constants, and x is the variable.

**What is a Perfect Square Trinomial?**

A perfect square trinomial is a quadratic expression that can be written in the form:

(x + d)^2 = x^2 + 2dx + d^2

where d is a constant.

**Factoring x^2 - 5x - 24 = 0**

Now, let's try to factor the given equation x^2 - 5x - 24 = 0.

**Step 1: Look for two numbers whose product is -24 and whose sum is -5.**

These numbers are -8 and 3, since (-8)(3) = -24 and (-8) + 3 = -5.

**Step 2: Rewrite the equation in the form (x + d)(x + e) = 0**

Using the numbers we found in Step 1, we can rewrite the equation as:

(x - 8)(x + 3) = 0

**Step 3: Solve for x**

To solve for x, we set each factor equal to 0 and solve for x:

x - 8 = 0 --> x = 8

x + 3 = 0 --> x = -3

Therefore, the solutions to the equation x^2 - 5x - 24 = 0 are x = 8 and x = -3.

**Conclusion**

In this article, we have seen how to factor a quadratic equation, specifically the equation x^2 - 5x - 24 = 0. We have identified the perfect square trinomial and used it to factor the equation, and finally, we have solved for x.

Remember, factoring quadratic equations is an important skill in algebra, and it can be used to solve a wide range of problems in mathematics, physics, and engineering.