Solving Quadratic Equations By Completing The Square Examples With Answers

Solving Quadratic Equations by Completing the Square: Examples with Answers

Quadratic equations, those pesky equations in the form ax² + bx + c = 0, can seem daunting. But fear not! One powerful method for solving them is completing the square. This technique allows us to manipulate the equation into a perfect square trinomial, making it easily solvable. This article will guide you through the process with clear examples and answers, helping you master this essential algebraic skill. We'll cover various scenarios, including those with leading coefficients other than 1.

What is Completing the Square?

Completing the square is a method used to manipulate a quadratic expression into a perfect square trinomial – a trinomial that can be factored into the square of a binomial. The general form of a perfect square trinomial is a² + 2ab + b² = (a + b)². The key is to identify the missing term needed to create this perfect square.

Steps to Completing the Square:

1. Ensure the leading coefficient is 1: If the coefficient of x² (a) is not 1, divide the entire equation by 'a'.
2. Move the constant term to the right side: Isolate the x² and x terms 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 maintains the balance of the equation.
4. Factor the perfect square trinomial: The left side should now factor into a perfect square (a + b)².
5. Solve for x: Take the square root of both sides, remembering to account for both positive and negative roots. Then, isolate x to find the solutions.

Examples with Detailed Solutions:

Let's work through some examples to solidify your understanding.

Example 1: Simple Case

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

1. Leading coefficient is 1.
2. Move the constant: x² + 6x = -5
3. Complete the square: Half of 6 is 3, and 3² = 9. Add 9 to both sides: x² + 6x + 9 = -5 + 9
4. Factor: (x + 3)² = 4
5. Solve for x: √(x + 3)² = ±√4 => x + 3 = ±2 => x = -3 ± 2 Therefore, x = -1 or x = -5

Example 2: Leading Coefficient Not Equal to 1

Solve 2x² - 8x + 6 = 0 by completing the square.

1. Divide by the leading coefficient: x² - 4x + 3 = 0
2. Move the constant: x² - 4x = -3
3. Complete the square: Half of -4 is -2, and (-2)² = 4. Add 4 to both sides: x² - 4x + 4 = -3 + 4
4. Factor: (x - 2)² = 1
5. Solve for x: √(x - 2)² = ±√1 => x - 2 = ±1 => x = 2 ± 1 Therefore, x = 1 or x = 3

Example 3: Fractional Coefficients

Solve x² + 5x + 2 = 0 by completing the square.

1. Leading coefficient is 1.
2. Move the constant: x² + 5x = -2
3. Complete the square: Half of 5 is 5/2, and (5/2)² = 25/4. Add 25/4 to both sides: x² + 5x + 25/4 = -2 + 25/4 = 17/4
4. Factor: (x + 5/2)² = 17/4
5. Solve for x: √(x + 5/2)² = ±√(17/4) => x + 5/2 = ±√17/2 => x = -5/2 ± √17/2

Conclusion:

Completing the square is a versatile method for solving quadratic equations, even those with fractional or non-unitary leading coefficients. By following these steps and practicing with various examples, you'll become proficient in this valuable algebraic technique. Remember to always check your answers by substituting them back into the original equation. Mastering completing the square will enhance your understanding of quadratic equations and prepare you for more advanced mathematical concepts.