Solving $x^2 + 4x = 45$ By Completing The Square A Step-by-Step Guide

Jul 15, 2025 by ADMIN 70 views

Solve Quadratic Equations by Completing the Square

In mathematics, quadratic equations play a crucial role. They appear in various fields, from physics to engineering, and understanding how to solve them is essential. One powerful method for solving quadratic equations is completing the square. This article provides a detailed guide on how to solve a quadratic equation using this technique, using the example $x^2 + 4x = 45$ .

Understanding Quadratic Equations

Before diving into the method of completing the square, it's important to grasp the basics of quadratic equations. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and x represents the unknown variable. The goal is to find the values of x that satisfy the equation, which are also known as the roots or solutions of the equation.

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful when the quadratic equation cannot be easily factored.

Why Completing the Square?

Completing the square might seem like a roundabout method, especially when compared to other techniques like factoring or using the quadratic formula. However, it offers several advantages:

Conceptual Understanding: It provides a deeper understanding of the structure of quadratic equations and how their roots are derived.
Derivation of the Quadratic Formula: The quadratic formula itself is derived using the method of completing the square. Understanding this method helps in appreciating the origin of the formula.
Solving Specific Types of Equations: Completing the square is particularly useful when the quadratic equation doesn't factor easily or when dealing with equations in vertex form.

Steps to Completing the Square

Now, let's outline the general steps involved in completing the square:

Rewrite the Equation: If the coefficient of $x^2$ (i.e., a) is not 1, divide the entire equation by a. This ensures that the leading coefficient is 1, which is necessary for completing the square.
Move the Constant Term: Move the constant term (c) to the right side of the equation. This isolates the terms with x on the left side.
Complete the Square: Take half of the coefficient of the x term (i.e., b), square it, and add the result to both sides of the equation. This step is the heart of the method, creating a perfect square trinomial on the left side.
Factor the Perfect Square Trinomial: Factor the left side of the equation as a squared binomial. This binomial will be of the form $(x + k)^2$ or $(x - k)^2$ , where k is half of the coefficient of the x term from the original equation.
Solve for x: Take the square root of both sides of the equation. Remember to consider both the positive and negative square roots. Finally, solve for x to find the solutions of the quadratic equation.

Solving $x^2 + 4x = 45$ by Completing the Square

Let's apply the method of completing the square to solve the given equation:

$x^2 + 4x = 45$

Step 1: Rewrite the Equation

In this case, the coefficient of $x^2$ is already 1, so we don't need to divide the equation. We can proceed to the next step.

Step 2: Move the Constant Term

The constant term is already on the right side of the equation, so we don't need to move anything. The equation remains:

$x^2 + 4x = 45$

Step 3: Complete the Square

The coefficient of the x term is 4. Half of 4 is 2, and squaring 2 gives us 4. So, we add 4 to both sides of the equation:

$x^2 + 4x + 4 = 45 + 4$

Simplifying, we get:

$x^2 + 4x + 4 = 49$

Step 4: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial. It can be factored as:

$(x + 2)^2 = 49$

This is because $(x + 2)(x + 2) = x^2 + 4x + 4$ .

Step 5: Solve for x

Now, we take the square root of both sides of the equation:

$\sqrt{(x + 2)^2} = \pm\sqrt{49}$

This gives us:

$x + 2 = \pm 7$

We have two cases to consider:

Case 1: $x + 2 = 7$

Subtracting 2 from both sides, we get:

$x = 7 - 2$

$x = 5$
Case 2: $x + 2 = -7$

Subtracting 2 from both sides, we get:

$x = -7 - 2$

$x = -9$

Therefore, the solutions to the quadratic equation $x^2 + 4x = 45$ are $x = 5$ and $x = -9$ .

Verifying the Solutions

To ensure our solutions are correct, we can substitute them back into the original equation:

For x = 5:

$(5)^2 + 4(5) = 25 + 20 = 45$

The equation holds true.

For x = -9:

$(-9)^2 + 4(-9) = 81 - 36 = 45$

The equation also holds true.

Both solutions satisfy the original equation, confirming that our answers are correct.

Applications of Completing the Square

Completing the square is not just a method for solving quadratic equations; it has several other important applications in mathematics and related fields. Here are a few notable examples:

1. Deriving the Quadratic Formula

The quadratic formula is a general solution for any quadratic equation of the form $ax^2 + bx + c = 0$ . It provides a direct way to find the roots of a quadratic equation without having to factor or complete the square each time. However, the quadratic formula itself is derived using the method of completing the square. By applying the steps of completing the square to the general form of the quadratic equation, we can arrive at the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Understanding the derivation of the quadratic formula through completing the square gives a deeper insight into why the formula works and its connection to the underlying structure of quadratic equations.

2. Converting to Vertex Form

The vertex form of a quadratic equation is a way of expressing the equation that reveals the vertex of the parabola represented by the equation. The vertex is the point where the parabola changes direction, either at its minimum (if the parabola opens upwards) or its maximum (if the parabola opens downwards). The vertex form of a quadratic equation is given by:

$y = a(x - h)^2 + k$

where (h, k) are the coordinates of the vertex. Completing the square can be used to convert a quadratic equation from its standard form ( $ax^2 + bx + c = 0$ ) to vertex form. This is done by completing the square for the x terms and rewriting the equation in the form above. The vertex form is particularly useful for graphing quadratic equations and for identifying the maximum or minimum value of the quadratic function.

3. Solving Optimization Problems

Quadratic equations and the concept of completing the square are often used in optimization problems, where the goal is to find the maximum or minimum value of a function. Many real-world scenarios can be modeled using quadratic functions, such as the trajectory of a projectile, the profit function of a business, or the cost function of a manufacturing process. By completing the square, we can rewrite the quadratic function in vertex form, which directly reveals the maximum or minimum value of the function and the value of x at which it occurs. This makes it possible to solve a wide range of optimization problems efficiently.

4. Applications in Calculus

In calculus, completing the square is a useful technique in various contexts, such as integration and finding limits. For example, when integrating certain types of rational functions, completing the square can help simplify the integrand and make the integration process easier. Similarly, when evaluating limits involving indeterminate forms, completing the square can sometimes help to rewrite the expression in a form that allows the limit to be computed more easily.

5. Applications in Physics and Engineering

Quadratic equations and the method of completing the square appear in numerous applications in physics and engineering. For example, in mechanics, the motion of a projectile under the influence of gravity can be described by a quadratic equation, and completing the square can be used to find the maximum height reached by the projectile and the time at which it occurs. In electrical engineering, quadratic equations are used to analyze circuits and determine the values of circuit components. In structural engineering, completing the square can be used to analyze the stability of structures and design them to withstand external forces.

Conclusion

Completing the square is a valuable technique for solving quadratic equations and understanding their properties. By following the steps outlined in this article, you can solve any quadratic equation using this method. It not only provides the solutions but also enhances your understanding of the structure and behavior of quadratic equations. Remember to practice and apply this method to various problems to master it. Whether you're a student learning algebra or a professional in a field that uses mathematical modeling, the ability to solve quadratic equations by completing the square is a powerful tool in your arsenal.

In addition to its direct application in solving quadratic equations, completing the square has broader implications in mathematics and related fields. From deriving the quadratic formula to converting equations to vertex form, solving optimization problems, and simplifying expressions in calculus, the technique offers a versatile approach to tackling mathematical challenges. Its presence in physics and engineering underscores its practical significance in real-world applications. Mastering completing the square enhances problem-solving skills, deepens mathematical insight, and provides a valuable tool for various disciplines.

By embracing completing the square as part of your mathematical toolkit, you'll be better equipped to handle quadratic equations and their diverse applications. Whether you're a student preparing for exams or a professional seeking efficient solutions, understanding and applying this technique will empower you to navigate the world of quadratics with confidence and expertise.