Completing the Square Method – Step-by-Step Guide with Examples

1. Introduction In this lesson, you will learn the completing the square method. This algebraic technique is crucial for solving quadratic equations that cannot be factored, converting quadratic functions into vertex form for easy graphing, and deriving the general quadratic formula.

2. Core Concept Explanation The goal of completing the square is to construct a perfect square trinomial on one side of the equation. A perfect square trinomial is in the form: x^2 + 2dx + d^2 = (x + d)^2

If we rewrite the quadratic equation ax^2 + bx + c = 0 in the form: (x + d)^2 = e We can solve for x by taking the square root of both sides.

3. Step-by-Step Method To solve ax^2 + bx + c = 0 by completing the square: 1. Divide all terms by a so that the coefficient of x^2 is exactly 1. 2. Move the constant term to the right side of the equation. 3. Identify the coefficient of the x term. Take half of it, square it, and add this value to both sides of the equation. 4. Factor the left side as a perfect square: (x + \frac{b}{2a})^2. 5. Take the square root of both sides, remembering to include the \pm sign on the right. 6. Solve the resulting linear equations for x.

4. Worked Examples * Example 1: Solve x^2 - 6x - 7 = 0. * Solution: 1. Coefficient of x^2 is 1. Move constant: x^2 - 6x = 7. 2. Half of the x coefficient is -6 / 2 = -3. Square it: (-3)^2 = 9. 3. Add 9 to both sides: x^2 - 6x + 9 = 7 + 9 \Rightarrow x^2 - 6x + 9 = 16. 4. Factor the left side: (x - 3)^2 = 16. 5. Take the square root: x - 3 = \pm 4. 6. Roots: x = 3 + 4 = 7 and x = 3 - 4 = -1. * Example 2: Solve 2x^2 + 5x - 3 = 0. * Solution: 1. Divide by 2: x^2 + \frac{5}{2}x - \frac{3}{2} = 0. 2. Move constant: x^2 + \frac{5}{2}x = \frac{3}{2}. 3. Half of \frac{5}{2} is \frac{5}{4}. Square it: (\frac{5}{4})^2 = \frac{25}{16}. 4. Add to both sides: x^2 + \frac{5}{2}x + \frac{25}{16} = \frac{3}{2} + \frac{25}{16}. 5. Simplify the right side: \frac{3}{2} + \frac{25}{16} = \frac{24}{16} + \frac{25}{16} = \frac{49}{16}. 6. Factor: (x + \frac{5}{4})^2 = \frac{49}{16}. 7. Square root: x + \frac{5}{4} = \pm \frac{7}{4}. 8. Roots: x = -\frac{5}{4} + \frac{7}{4} = 0.5 and x = -\frac{5}{4} - \frac{7}{4} = -3. * Example 3: Convert x^2 - 4x + 7 to vertex form y = a(x - h)^2 + k. * Solution: 1. Group the x terms: (x^2 - 4x) + 7. 2. Complete the square inside: half of -4 is -2, squared is 4. 3. Add and subtract 4: (x^2 - 4x + 4 - 4) + 7 \Rightarrow (x - 2)^2 - 4 + 7 \Rightarrow (x - 2)^2 + 3. 4. The vertex form is y = (x - 2)^2 + 3.

5. Exam Relevance * Example WAEC Question: Solve the equation x^2 - 8x + 12 = 0 by completing the square method. * Solution: Move the constant: x^2 - 8x = -12. Add ( -8 / 2 )^2 = 16 to both sides: x^2 - 8x + 16 = -12 + 16 \Rightarrow (x - 4)^2 = 4 Take square root: x - 4 = \pm 2. Roots: x = 4 + 2 = 6 or x = 4 - 2 = 2.

6. Common Mistakes * Not dividing by a first: Forgetting to divide all terms by the coefficient of x^2 before adding the squared term. * Adding to only one side: Adding the squared term to the left side of the equation but forgetting to add it to the right side.

7. Practice Questions 1. Solve x^2 + 4x - 5 = 0 by completing the square. * Answer: x = 1 or x = -5. 2. Write x^2 + 8x + 11 = 0 in the form (x+d)^2 = e. * Answer: (x + 4)^2 = 5.

8. Summary / Cheat Sheet * Perfect Square Identity: x^2 + 2dx + d^2 = (x+d)^2. * Key Step: Add (\frac{b}{2})^2 to both sides (after making a=1). * Vertex form y = a(x-h)^2 + k reveals the vertex (h, k) directly.