Completing The Square Examples And Answers

Completing the Square: Examples and Answers

Completing the square is a crucial algebraic technique used extensively in various mathematical fields, from solving quadratic equations to simplifying complex expressions and even in calculus when dealing with integrals. Understanding this method is vital for mastering higher-level mathematics. This comprehensive guide will provide you with a step-by-step approach to completing the square, numerous examples with detailed solutions, and various applications to solidify your understanding.

Understanding the Concept

The core idea behind completing the square lies in manipulating a quadratic expression of the form ax² + bx + c into a perfect square trinomial, which can then be easily factored. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as (x + p)². This process involves adding and subtracting a specific value to the original expression.

The key formula to remember is:

(x + a)² = x² + 2ax + a²

Notice how the constant term, a², is the square of half the coefficient of the x term (2a). This is the foundation of the completing the square method.

Step-by-Step Guide to Completing the Square

Let's break down the process into manageable steps:

1. Ensure the coefficient of x² is 1: If the coefficient of x² (a) is not 1, divide the entire equation by 'a'. This ensures that the process remains consistent and simplifies calculations.

2. Move the constant term to the other side: Isolate the terms containing x² and x on one side of the equation, leaving the constant term (c) on the other.

3. Find half the coefficient of x and square it: Take half of the coefficient of x (b/2) and square it ((b/2)²). This is the value you'll add to both sides of the equation to "complete the square."

4. Add the value to both sides: Add (b/2)² to both sides of the equation to maintain balance. Remember, what you add to one side, you must add to the other.

5. Factor the perfect square trinomial: The left-hand side of the equation should now be a perfect square trinomial, which can be factored into the square of a binomial (x + b/2)².

6. Solve for x (if applicable): If you're solving a quadratic equation, you can now take the square root of both sides and solve for x.

Examples with Detailed Solutions

Let's work through several examples to illustrate the process.

Example 1: Simple Quadratic

Solve x² + 6x + 5 = 0 using completing the square.

1. Coefficient of x² is already 1.

2. Move the constant: x² + 6x = -5

3. Half the coefficient of x and square it: (6/2)² = 9

4. Add to both sides: x² + 6x + 9 = -5 + 9 => x² + 6x + 9 = 4

5. Factor the perfect square: (x + 3)² = 4

6. Solve for x: Taking the square root of both sides: x + 3 = ±2. Therefore, x = -1 or x = -5.

Example 2: Coefficient of x² is not 1

Solve 2x² - 8x + 5 = 0 using completing the square.

1. Divide by the coefficient of x²: x² - 4x + 5/2 = 0

2. Move the constant: x² - 4x = -5/2

3. Half the coefficient of x and square it: (-4/2)² = 4

4. Add to both sides: x² - 4x + 4 = -5/2 + 4 => x² - 4x + 4 = 3/2

5. Factor the perfect square: (x - 2)² = 3/2

6. Solve for x: x - 2 = ±√(3/2) => x = 2 ± √(3/2)

Example 3: Leading Coefficient is Negative

Solve -x² + 4x - 2 = 0 using completing the square.

1. Multiply by -1 to make the leading coefficient positive: x² - 4x + 2 = 0

2. Move the constant: x² - 4x = -2

3. Half the coefficient of x and square it: (-4/2)² = 4

4. Add to both sides: x² - 4x + 4 = -2 + 4 => x² - 4x + 4 = 2

5. Factor the perfect square: (x - 2)² = 2

6. Solve for x: x - 2 = ±√2 => x = 2 ± √2

Example 4: Completing the Square to Find the Vertex of a Parabola

Find the vertex of the parabola represented by the equation y = x² - 10x + 21.

We complete the square to rewrite the equation in vertex form, y = a(x - h)² + k, where (h, k) is the vertex.

1. The coefficient of x² is already 1.

2. Group the x terms: y = (x² - 10x) + 21

3. Half the coefficient of x and square it: (-10/2)² = 25

4. Add and subtract 25: y = (x² - 10x + 25 - 25) + 21

5. Factor the perfect square and simplify: y = (x - 5)² - 25 + 21 => y = (x - 5)² - 4

The vertex of the parabola is (5, -4).

Advanced Applications of Completing the Square

Completing the square isn't limited to solving quadratic equations. It extends to various other applications:

• Derivation of the Quadratic Formula: The quadratic formula itself is derived using the method of completing the square on the general quadratic equation ax² + bx + c = 0.

• Graphing Parabolas: Completing the square helps convert the standard form of a parabola's equation into vertex form, making it easy to identify the vertex, axis of symmetry, and direction of opening.

• Calculus: Completing the square is used in integral calculus to simplify integrals involving quadratic expressions in the denominator. This allows for the use of trigonometric substitutions or other techniques to solve the integral.

• Conic Sections: Completing the square plays a vital role in identifying and graphing conic sections (circles, ellipses, parabolas, and hyperbolas) by converting their equations into standard forms.

Practice Problems

To reinforce your understanding, try completing the square for these quadratic equations:

1. x² + 8x + 12 = 0
2. 3x² - 6x + 2 = 0
3. -2x² + 12x - 10 = 0
4. x² - 5x + 6 = 0
5. 4x² + 12x + 5 = 0

Conclusion

Completing the square is a powerful algebraic technique with wide-ranging applications. While initially appearing complex, mastering this method through consistent practice unlocks a deeper understanding of quadratic equations and their behavior. By following the step-by-step guide and working through the examples provided, you'll gain the confidence to tackle more challenging problems and appreciate the versatility of this essential mathematical tool. Remember to practice regularly and explore different types of problems to fully grasp its applications. This will greatly benefit your mathematical studies and problem-solving skills in various contexts.