Solving Quadratic Equations By Completing The Square Examples With Answers

Solving Quadratic Equations by Completing the Square: Examples with Answers

Quadratic equations, those pesky polynomial expressions of degree two (like ax² + bx + c = 0), often present a challenge. While the quadratic formula provides a direct solution, the method of completing the square offers a deeper understanding of the underlying structure of quadratic equations and serves as a foundation for more advanced mathematical concepts. This comprehensive guide will delve into the intricacies of completing the square, providing step-by-step examples and detailed explanations, ensuring you master this valuable technique.

Understanding the Concept of Completing the Square

Before we dive into examples, let's establish a solid understanding of the core principle. Completing the square involves manipulating a quadratic equation to transform it into a perfect square trinomial—a trinomial that can be factored into the square of a binomial. This process relies heavily on the algebraic identity:

(a + b)² = a² + 2ab + b²

The goal is to rewrite the equation in the form (x + p)² = q, where p and q are constants. From this form, solving for x becomes relatively straightforward.

The Key Steps: A Systematic Approach

The process of completing the square involves several key steps:

1. Ensure the coefficient of x² is 1: If the coefficient of x² is not 1, divide the entire equation by that coefficient.

2. Move the constant term to the right side: Isolate the terms containing x on the left side of the equation.

3. Find the value to "complete the square": Take half of the coefficient of x (b/2), square it ((b/2)²), and add it to both sides of the equation. This crucial step creates the perfect square trinomial on the left side.

4. Factor the perfect square trinomial: Rewrite the left side as the square of a binomial (x + b/2)².

5. Solve for x: Take the square root of both sides, remembering to account for both positive and negative square roots. Then, isolate x to find the solutions.

Examples of Completing the Square: From Simple to Complex

Let's work through several examples, progressively increasing in complexity, to solidify your understanding.

Example 1: A Simple Quadratic Equation

Solve the equation: x² + 6x + 5 = 0

Step 1: The coefficient of x² is already 1.

Step 2: Move the constant term to the right side: x² + 6x = -5

Step 3: Find the value to complete the square: (6/2)² = 9. Add 9 to both sides: x² + 6x + 9 = -5 + 9

Step 4: Factor the perfect square trinomial: (x + 3)² = 4

Step 5: Solve for x: x + 3 = ±√4 => x + 3 = ±2 => x = -3 ± 2. Therefore, x = -1 or x = -5.

Example 2: A Quadratic Equation with a Leading Coefficient Greater Than 1

Solve the equation: 2x² + 8x - 10 = 0

Step 1: Divide the entire equation by 2: x² + 4x - 5 = 0

Step 2: Move the constant term: x² + 4x = 5

Step 3: Complete the square: (4/2)² = 4. Add 4 to both sides: x² + 4x + 4 = 5 + 4

Step 4: Factor: (x + 2)² = 9

Step 5: Solve: x + 2 = ±√9 => x + 2 = ±3 => x = -2 ± 3. Therefore, x = 1 or x = -5.

Example 3: A Quadratic Equation with a Fractional Coefficient of x

Solve the equation: x² + 3x + 1/4 = 0

Step 1: The coefficient of x² is already 1.

Step 2: Move the constant term: x² + 3x = -1/4

Step 3: Complete the square: (3/2)² = 9/4. Add 9/4 to both sides: x² + 3x + 9/4 = -1/4 + 9/4

Step 4: Factor: (x + 3/2)² = 8/4 = 2

Step 5: Solve: x + 3/2 = ±√2 => x = -3/2 ± √2

Example 4: A Quadratic Equation with a Negative Coefficient of x

Solve the equation: x² - 5x + 6 = 0

Step 1: The coefficient of x² is already 1.

Step 2: Move the constant term: x² - 5x = -6

Step 3: Complete the square: (-5/2)² = 25/4. Add 25/4 to both sides: x² - 5x + 25/4 = -6 + 25/4 = 1/4

Step 4: Factor: (x - 5/2)² = 1/4

Step 5: Solve: x - 5/2 = ±√(1/4) => x - 5/2 = ±1/2 => x = 5/2 ± 1/2. Therefore, x = 3 or x = 2

Dealing with Complex Numbers: Expanding the Scope

Completing the square also works when the resulting equation involves complex numbers. Let's examine an example:

Example 5: A Quadratic Equation Resulting in Complex Roots

Solve the equation: x² + 2x + 5 = 0

Step 1: The coefficient of x² is already 1.

Step 2: Move the constant term: x² + 2x = -5

Step 3: Complete the square: (2/2)² = 1. Add 1 to both sides: x² + 2x + 1 = -5 + 1

Step 4: Factor: (x + 1)² = -4

Step 5: Solve: x + 1 = ±√(-4) => x + 1 = ±2i => x = -1 ± 2i. The solutions are complex numbers.

The Significance of Completing the Square: Beyond Solving Equations

While primarily used to solve quadratic equations, completing the square holds broader significance:

• Deriving the Quadratic Formula: The quadratic formula itself is derived using the method of completing the square. Understanding this derivation deepens your appreciation of the formula's origins.

• Graphing Parabolas: Completing the square is essential for finding the vertex of a parabola, represented by the quadratic equation. The vertex form of a quadratic equation, (x - h)² + k, where (h, k) is the vertex, is directly obtained through completing the square. This allows for accurate graphing of the parabola.

• Calculus Applications: Completing the square plays a crucial role in various calculus applications, including integration and solving differential equations. It's frequently used to simplify complex expressions and facilitate calculations.

Conclusion: Mastering a Fundamental Mathematical Technique

Completing the square is more than just a method for solving quadratic equations; it's a fundamental algebraic technique with far-reaching implications in mathematics. By mastering this technique, you gain a deeper understanding of quadratic equations, their structure, and their solutions. The examples provided, ranging from straightforward to more complex scenarios, illustrate the versatility of this powerful method and equip you to confidently tackle quadratic equations in various contexts. Remember to practice regularly to reinforce your understanding and build your problem-solving skills. The more you practice, the more intuitive and efficient this method will become.