How To Do Complete The Square

Completing the square is a powerful algebraic technique used to rewrite a quadratic expression in a more convenient form. This technique is invaluable for solving quadratic equations, graphing parabolas, and simplifying complex algebraic expressions. Understanding and mastering completing the square unlocks a deeper understanding of quadratic functions and their applications.

Understanding Quadratic Expressions

Before diving into the steps of completing the square, it's crucial to understand the basic form of a quadratic expression and its components. A quadratic expression is generally written as:

ax² + bx + c

Where:

• a, b, and c are constants (numbers), and a ≠ 0
• x is the variable

The goal of completing the square is to transform this expression into the form:

a(x + h)² + k

Where h and k are also constants. This form is particularly useful because it reveals the vertex of the parabola represented by the quadratic equation and makes it easier to solve for x.

Why Complete the Square?

Completing the square offers several advantages:

• Solving Quadratic Equations: It provides a reliable method for solving any quadratic equation, even those that are difficult or impossible to factor.
• Finding the Vertex of a Parabola: The completed square form directly reveals the vertex (h, k) of the parabola represented by the quadratic equation. The vertex is the point where the parabola reaches its maximum or minimum value.
• Graphing Quadratic Functions: Knowing the vertex and the shape of the parabola (determined by the coefficient a) makes graphing quadratic functions straightforward.
• Integration in Calculus: Completing the square can simplify integrands in calculus, making them easier to integrate.

The Steps to Completing the Square

Let's break down the process of completing the square step-by-step with examples. We'll start with the simplest case where a = 1, and then move on to cases where a ≠ 1.

Case 1: a = 1 (Leading Coefficient is 1)

This is the most straightforward scenario. Consider the quadratic expression:

x² + 6x + 5

Step 1: Group the x² and x terms:

Isolate the terms containing x² and x, leaving the constant term aside for now.

(x² + 6x) + 5

Step 2: Complete the Square Inside the Parentheses:

• Take half of the coefficient of the x term (which is 6 in this case). Half of 6 is 3.
• Square the result from the previous step. 3 squared is 9.
• Add and subtract this value inside the parentheses. This crucial step ensures that you're not changing the overall value of the expression.

(x² + 6x + 9 - 9) + 5

Step 3: Factor the Perfect Square Trinomial:

The expression inside the parentheses, x² + 6x + 9, is now a perfect square trinomial. It can be factored as (x + 3)².

((x + 3)² - 9) + 5

Step 4: Simplify:

Remove the parentheses and combine the constant terms.

(x + 3)² - 9 + 5

(x + 3)² - 4

The expression is now in completed square form: (x + 3)² - 4.

Example 2: x² - 8x + 12

1. Group terms: (x² - 8x) + 12
2. Complete the square: Half of -8 is -4, and (-4)² is 16. (x² - 8x + 16 - 16) + 12
3. Factor: ((x - 4)² - 16) + 12
4. Simplify: (x - 4)² - 16 + 12 = (x - 4)² - 4

Case 2: a ≠ 1 (Leading Coefficient is Not 1)

When the coefficient of the x² term ( a ) is not 1, an extra step is required. Consider the quadratic expression:

2x² + 8x + 6

Step 1: Factor out the Leading Coefficient from the x² and x terms:

Factor out the coefficient a (in this case, 2) from the terms containing x² and x.

2(x² + 4x) + 6

Step 2: Complete the Square Inside the Parentheses:

Focus on the expression inside the parentheses: (x² + 4x).

• Take half of the coefficient of the x term (which is 4). Half of 4 is 2.
• Square the result from the previous step. 2 squared is 4.
• Add and subtract this value inside the parentheses.

2(x² + 4x + 4 - 4) + 6

Step 3: Factor the Perfect Square Trinomial:

Factor the perfect square trinomial inside the parentheses.

2((x + 2)² - 4) + 6

Step 4: Distribute the Leading Coefficient:

Distribute the factored-out coefficient (2 in this case) to both terms inside the parentheses.

2(x + 2)² - 8 + 6

Step 5: Simplify:

Combine the constant terms.

2(x + 2)² - 2

The expression is now in completed square form: 2(x + 2)² - 2.

Example 2: 3x² - 12x + 5

1. Factor out the leading coefficient: 3(x² - 4x) + 5
2. Complete the square: Half of -4 is -2, and (-2)² is 4. 3(x² - 4x + 4 - 4) + 5
3. Factor: 3((x - 2)² - 4) + 5
4. Distribute: 3(x - 2)² - 12 + 5
5. Simplify: 3(x - 2)² - 7

Solving Quadratic Equations by Completing the Square

Completing the square is a powerful technique for solving quadratic equations. Let's illustrate this with an example.

Example: Solve x² + 6x + 5 = 0

1. Complete the square: We already completed the square for this expression earlier: (x + 3)² - 4 = 0

2. Isolate the squared term: Add 4 to both sides of the equation. (x + 3)² = 4

3. Take the square root of both sides: Remember to consider both the positive and negative square roots. x + 3 = ±2

4. Solve for x: Subtract 3 from both sides. x = -3 ± 2

   This gives us two solutions:

   • x = -3 + 2 = -1
   • x = -3 - 2 = -5

Therefore, the solutions to the quadratic equation x² + 6x + 5 = 0 are x = -1 and x = -5.

Example 2: Solve 2x² + 8x + 6 = 0

1. Complete the square: We already completed the square earlier: 2(x + 2)² - 2 = 0

2. Isolate the squared term: Add 2 to both sides and then divide by 2: (x + 2)² = 1

3. Take the square root of both sides: x + 2 = ±1

4. Solve for x: Subtract 2 from both sides: x = -2 ± 1

   This gives us two solutions:

   • x = -2 + 1 = -1
   • x = -2 - 1 = -3

Finding the Vertex of a Parabola

The completed square form directly reveals the vertex of the parabola represented by the quadratic equation. Recall that the completed square form is:

a(x + h)² + k

The vertex of the parabola is the point (-h, k). Note the negative sign in front of h.

Example 1: Find the vertex of the parabola represented by y = (x + 3)² - 4

In this case, h = -3 and k = -4. Therefore, the vertex of the parabola is (-3, -4).

Example 2: Find the vertex of the parabola represented by y = 2(x + 2)² - 2

Here, h = -2 and k = -2. Thus, the vertex is (-2, -2).

Example 3: Find the vertex of the parabola represented by y = x² - 6x + 11

1. Complete the square:

   • Group terms: (x² - 6x) + 11
   • Complete the square: Half of -6 is -3, and (-3)² is 9. (x² - 6x + 9 - 9) + 11
   • Factor: ((x - 3)² - 9) + 11
   • Simplify: (x - 3)² + 2

2. Identify the vertex: Now the equation is in the form y = (x - 3)² + 2. Therefore, h = 3 and k = 2. The vertex is (3, 2).

Common Mistakes to Avoid

• Forgetting to Add and Subtract: When completing the square, it's crucial to both add and subtract the squared value inside the parentheses. Adding only would change the value of the expression.
• Incorrectly Factoring the Leading Coefficient: When a ≠ 1, make sure to factor it out correctly from both the x² and x terms.
• Forgetting to Distribute: After completing the square inside the parentheses when a ≠ 1, remember to distribute the a value back into the parentheses.
• Sign Errors: Pay close attention to signs, especially when dealing with negative coefficients.
• Incorrectly Identifying the Vertex: Remember that the vertex is (-h, k), not (h, k). The sign of the h value is flipped when identifying the vertex.

Applications of Completing the Square

Beyond solving quadratic equations and finding vertices, completing the square has applications in various areas of mathematics and physics:

• Calculus: As mentioned earlier, completing the square can simplify integrands, making integration easier.
• Conic Sections: It's used to rewrite the equations of conic sections (circles, ellipses, hyperbolas, parabolas) in standard form, which makes it easier to identify their properties.
• Optimization Problems: Completing the square can be used to find the maximum or minimum value of a quadratic function, which is useful in optimization problems.
• Physics: It appears in various physics problems involving projectile motion, energy, and other areas where quadratic relationships arise.

Conclusion

Completing the square is a fundamental algebraic technique with wide-ranging applications. By mastering this technique, you gain a deeper understanding of quadratic functions, their properties, and their role in various mathematical and scientific contexts. Practice is key to becoming proficient in completing the square. Work through numerous examples, paying close attention to each step, and you'll soon find yourself confidently applying this powerful tool to solve a variety of problems.