A quadratic equation is a second-degree polynomial equation in a single variable. It can be represented in the form **ax^2 + bx + c = 0**, where *a*, *b*, and *c* are constants.

## Introduction to Quadratic Equations

**A quadratic equation** is a polynomial equation of degree 2. It represents a parabolic curve when graphed. The general form of a quadratic equation is **ax^2 + bx + c = 0**, where *a*, *b*, and *c* are constants, and *x* is the variable.

In this article, we will focus on solving the quadratic equation **x^2 + 10x + 21 = 0**. By understanding the steps involved in solving this equation, we can apply the same principles to solve other quadratic equations as well.

## Factoring the Quadratic Equation

The first method we will explore to solve the quadratic equation **x^2 + 10x + 21 = 0** is **factoring**. Factoring involves finding two binomials that, when multiplied, result in the given quadratic equation.

Let’s factor the given quadratic equation:

**x^2 + 10x + 21 = 0**

**(x + 7)(x + 3) = 0**

By setting each binomial equal to zero, we can find the values of *x* that satisfy the equation:

**x + 7 = 0** or **x + 3 = 0**

Solving these equations, we find that **x = -7** or **x = -3**. Hence, the solutions to the quadratic equation **x^2 + 10x + 21 = 0** are **x = -7** and **x = -3**.

## Using the Quadratic Formula

The quadratic formula is another method to solve quadratic equations. It is particularly useful when factoring is not easily achievable or when dealing with more complex equations.

The quadratic formula states that for any quadratic equation **ax^2 + bx + c = 0**, the solutions can be found using the formula:

**x = (-b ± √(b^2 – 4ac)) / 2a**

Applying the quadratic formula to the equation **x^2 + 10x + 21 = 0**, we have:

**x = (-10 ± √(10^2 – 4*1*21)) / 2*1**

**x = (-10 ± √(100 – 84)) / 2**

**x = (-10 ± √16) / 2**

**x = (-10 ± 4) / 2**

Simplifying further, we get:

**x = (-10 + 4) / 2** or **x = (-10 – 4) / 2**

**x = -6 / 2** or **x = -14 / 2**

**x = -3** or **x = -7**

Therefore, the solutions to the quadratic equation **x^2 + 10x + 21 = 0** are **x = -3** and **x = -7**.

## Graphical Representation

Graphing the quadratic equation **x^2 + 10x + 21 = 0** allows us to visually understand its solutions. By plotting the equation on a graph, we can determine where the curve intersects the x-axis, representing the values of *x* that satisfy the equation.

The graph of the quadratic equation **x^2 + 10x + 21 = 0** is a downward-opening parabola. It intersects the x-axis at the points **x = -7** and **x = -3**, confirming our earlier solutions.

## Real-Life Applications

Quadratic equations have numerous real-life applications, especially in fields such as physics, engineering, and finance. Some common examples include:

- Projectile motion: Quadratic equations are used to model the path of projectiles, such as objects thrown or launched into the air.
- Optimization problems: Quadratic equations help solve optimization problems, where the goal is to find the maximum or minimum value of a certain quantity.
- Financial modeling: Quadratic equations are used in finance to calculate the value of investments, options, and risk assessments.

## Conclusion

Solving quadratic equations is an essential skill in mathematics. In this article, we explored the quadratic equation **x^2 + 10x + 21 = 0** and demonstrated two methods to find its solutions: factoring and using the quadratic formula. By understanding these techniques, we can apply them to solve other quadratic equations and tackle real-life problems in various fields.