Intermediate Steps in Completing the Square for Quadratic Equations

Introduction

Completing the square is a powerful algebraic technique used to solve quadratic equations. This method transforms a quadratic equation into a perfect square trinomial, making the solution process more straightforward. In this article, we will explore the intermediate steps involved in completing the square, using specific examples to illustrate the process.

Understanding the Process: The Standard Quadratic Form

The standard form of a quadratic equation is given by:

x2 bx c 0

We will focus on the intermediate steps involved in completing the square, particularly in transforming a quadratic equation into the form x - a2 b.

Example 1: -2x2 - 41x - 106 -13x

Let's start with the equation -2x2 - 41x - 106 -13x.

Rearrange the equation: -2x2 - 41x - 106 13x 0 This simplifies to: -2x2 - 28x - 106 0 Factor out -2 from the quadratic terms: -2x2 - 14x - 106 0 Move the constant term to the right side: -2x2 - 14x 106 Divide by -2: x2 7x -53 Complete the square: Take half of the coefficient of x, which is 7, square it, and add it to both sides. Half of 7 is 3.5, and squaring it gives 12.25. x2 7x 12.25 -53 12.25 This simplifies to: (x 3.5)2 -40.75 The intermediate step in the form x - a2 b is: (x 3.5)2 -40.75.

Example 2: 2x2 - 9x 7x - 64

First, let's combine like terms: 2x2 - 9x - 7x 64 0 This simplifies to: 2x2 - 16x 64 0 Rearrange the equation: 2x2 - 16x -64 Divide all terms by 2: x2 - 8x -32 Complete the square: To complete the square, take half of the coefficient of x, which is -8, square it (42 16), and add it to both sides of the equation. x2 - 8x 16 -32 16 This simplifies to: (x - 4)2 -16 The intermediate step in the form x - a2 b is: (x - 4)2 -16.

Example 3: -2x2 - 22x - 3 -13

Rearrange the equation: -2x2 - 22x - 3 13 0 This simplifies to: -2x2 - 22x 10 0 Divide by -2: x2 11x - 5 0 Move the constant term to the right side: x2 11x 5 Complete the square: Take half of the coefficient of x, which is 11, square it (5.52 30.25), and add it to both sides of the equation. x2 11x 30.25 5 30.25 This simplifies to: (x 5.5)2 35.25 The intermediate step in the form x - a2 b is: (x 5.5)2 35.25.

Conclusion

Completing the square is a valuable algebraic technique that provides a structured way to solve quadratic equations. The intermediate steps, like transforming the equation into the form x - a2 b, are crucial and form the basis for finding the roots of the equation. By following these intermediate steps, one can systematically work through the process and arrive at the final solution.