Solving Quadratic Equations By Completing The Square Step-by-Step

In the realm of algebra, quadratic equations hold a prominent position, frequently encountered in various mathematical and scientific contexts. Among the arsenal of techniques available for tackling these equations, completing the square stands out as a particularly insightful and versatile method. This article delves into the intricacies of completing the square, providing a comprehensive guide to solving quadratic equations using this approach. We will dissect the steps involved, elucidate the underlying principles, and illustrate the process with a concrete example, ensuring a thorough understanding of this powerful technique.

Understanding Quadratic Equations and the Method of Completing the Square

At its core, a quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as its roots or zeros, represent the values of x that satisfy the equation. These roots can be real or complex numbers, and their nature is determined by the discriminant, a key component of the quadratic formula.

The Power of Completing the Square

Completing the square is a method that transforms a quadratic equation into a perfect square trinomial, a trinomial that can be factored into the square of a binomial. This transformation allows us to isolate the variable x and solve for its values. The technique is not only useful for finding solutions but also provides a deeper understanding of the structure and properties of quadratic equations. It forms the basis for deriving the quadratic formula, a universal solution for quadratic equations.

Step-by-Step Guide to Completing the Square

To effectively solve a quadratic equation by completing the square, we follow a systematic series of steps. Let's break down the process into manageable stages:

1. Rearrange the Equation

Begin by rearranging the equation so that the constant term c is isolated on one side of the equation. This involves moving the constant term to the right side, leaving the terms containing x² and x on the left side. This step sets the stage for manipulating the equation into the desired form.

In our example, the given equation is:

$0 = -2x^2 + 2x + 3$

Subtracting 3 from both sides, we get:

$-3 = -2x^2 + 2x$

2. Factor out the Leading Coefficient

If the coefficient of the x² term (a) is not equal to 1, factor it out from the terms containing x² and x. This ensures that the coefficient of x² inside the parentheses is 1, a crucial requirement for completing the square. Factoring out the leading coefficient simplifies the subsequent steps and allows us to focus on manipulating the expression within the parentheses.

In our example, the coefficient of x² is -2. Factoring it out, we have:

$-3 = -2(x^2 - x)$

3. Complete the Square

This is the heart of the method. Take half of the coefficient of the x term (inside the parentheses), square it, and add it inside the parentheses. This step creates a perfect square trinomial, which can be factored into the square of a binomial. However, to maintain the balance of the equation, we must also add the same value to the other side. Note that since we factored out a coefficient in the previous step, we need to multiply the value we added inside the parentheses by that coefficient before adding it to the other side.

The coefficient of the x term inside the parentheses is -1. Half of -1 is -1/2, and squaring it gives us 1/4. Therefore, we add 1/4 inside the parentheses:

$-3 + ? = -2(x^2 - x + 1/4)$

Since we added 1/4 inside the parentheses, which is being multiplied by -2, we must add (-2) * (1/4) = -1/2 to the left side of the equation:

$-3 + (-1/2) = -2(x^2 - x + 1/4)$

This simplifies to:

$-3 - 1/2 = -2(x^2 - x + 1/4)$

4. Factor the Perfect Square Trinomial

The expression inside the parentheses is now a perfect square trinomial, which can be factored into the square of a binomial. The binomial will consist of x plus or minus half of the coefficient of the x term from the previous step. This factorization is the key to simplifying the equation and isolating x.

In our example, the perfect square trinomial is x² - x + 1/4, which factors into (x - 1/2)²:

$-3 - 1/2 = -2(x - 1/2)^2$

5. Isolate the Squared Term

Divide both sides of the equation by the coefficient that was factored out in Step 2. This isolates the squared term, bringing us closer to solving for x. Isolating the squared term is a critical step in the process, as it allows us to apply the square root property in the next step.

In our example, we divide both sides by -2:

(-3 - 1/2) / -2 = (x - 1/2)²

Simplifying the left side, we get:

7/4 = (x - 1/2)²

6. Take the Square Root

Take the square root of both sides of the equation. Remember to include both the positive and negative square roots, as both will satisfy the equation. This step introduces two possible solutions, reflecting the quadratic nature of the equation. The square root property is a fundamental tool in solving equations involving squared terms.

Taking the square root of both sides, we get:

±√(7/4) = x - 1/2

This simplifies to:

±√7 / 2 = x - 1/2

7. Solve for x

Isolate x by adding the constant term that was subtracted in the binomial to both sides of the equation. This gives us the two possible solutions for x. Solving for x involves a simple algebraic manipulation, but it is the final step in determining the roots of the quadratic equation.

Adding 1/2 to both sides, we get:

x = 1/2 ± √7 / 2

Therefore, the two solutions are:

x = (1 + √7) / 2 and x = (1 - √7) / 2

Applying the Steps to the Given Equation

Let's revisit the original equation and the initial steps taken:

$0 = -2x^2 + 2x + 3$

1. $-3 = -2x^2 + 2x$ (Subtracting 3 from both sides)
2. $-3 = -2(x^2 - x)$ (Factoring out -2)

The Next Crucial Step

As we've outlined in the step-by-step guide, the next critical step is to complete the square. This involves determining the value to add inside the parentheses to create a perfect square trinomial. As calculated earlier, we need to add (1/2)² = 1/4 inside the parentheses. However, since we factored out a -2, we must add (-2) * (1/4) = -1/2 to the left side of the equation.

Therefore, the next step is:

$-3 + (-1/2) = -2(x^2 - x + 1/4)$

This corresponds to step 3 in the original sequence, but with the correct value added to both sides.

Why Completing the Square Matters

Completing the square is more than just a technique for solving quadratic equations; it's a foundational concept that underpins many areas of mathematics. Here's why it's so important:

Deriving the Quadratic Formula

The quadratic formula, a cornerstone of algebra, is directly derived from the process of completing the square. By applying the steps of completing the square to the general quadratic equation ax² + bx + c = 0, we can arrive at the quadratic formula:

$x = (-b ± √(b² - 4ac)) / (2a)$

This demonstrates the fundamental connection between completing the square and the quadratic formula, highlighting the power of the technique.

Understanding the Vertex Form of a Quadratic Equation

Completing the square allows us to rewrite a quadratic equation in vertex form, which provides valuable information about the parabola represented by the equation. The vertex form is given by:

$y = a(x - h)² + k$

where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex, axis of symmetry, and direction of opening of the parabola, making it a valuable tool in graphing and analyzing quadratic functions.

Solving Optimization Problems

Completing the square can be used to solve optimization problems involving quadratic functions. By rewriting the function in vertex form, we can easily identify the maximum or minimum value of the function, which corresponds to the y-coordinate of the vertex. This technique is widely used in various fields, such as engineering, economics, and physics.

A Versatile Algebraic Technique

Beyond solving quadratic equations, completing the square is a versatile algebraic technique that can be applied in various contexts. It can be used to simplify expressions, solve inequalities, and even integrate certain functions in calculus. Its broad applicability makes it a valuable tool in the mathematician's arsenal.

Conclusion: Mastering Completing the Square

Completing the square is a powerful and versatile technique for solving quadratic equations and understanding their properties. By mastering the steps outlined in this guide, you can confidently tackle a wide range of quadratic equations and gain a deeper appreciation for the underlying mathematical principles. From deriving the quadratic formula to understanding the vertex form of a parabola, completing the square provides a valuable foundation for further exploration in mathematics and related fields. So, embrace this technique, practice its application, and unlock the power it holds for solving problems and expanding your mathematical understanding.