How to Use Completing the Square Technique to Simplify Quadratic Equations: A Step-by-Step Guide

Completing the Square

a process used to make a quadratic expression into a perfect square trinomial

Completing the square is a useful mathematical technique that is used to simplify and manipulate algebraic expressions. The primary use of completing the square is to transform a quadratic equation from the standard form, such as ax^2 + bx + c = 0, to a vertex form, such as a(x – h)^2 + k, where (h, k) is the vertex of the parabola.

To use the completing the square technique, follow these steps:

1. Make sure the coefficient of the x^2 term is 1. If not, divide the entire equation by the coefficient to make it 1.

2. Move the constant term to the right side of the equation.

3. Then divide the coefficient of the x-term by 2, square it, and add it to both sides of the equation.

4. On the left side, write the squared quantity as (x + b/2)^2.

5. Simplify the right side of the equation as much as possible and rewrite the equation in vertex form.

6. Finally, solve for x by taking the square root of both sides, if necessary.

Let’s illustrate the process with an example.

Suppose we want to complete the square of the quadratic equation 2x^2 + 8x + 1 = 0.

1. Since the coefficient of x^2 term is already 1, we can move directly to step 2.

2. Subtracting 1 from both sides, we get 2x^2 + 8x = -1.

3. Divide the coefficient of x-term by 2 and square it: (8/2)^2 = 16. So, we add 16 to both sides of the equation: 2x^2 + 8x + 16 = 15.

4. On the left side, we have (x + 2)^2 = 15/2.

5. Rewriting the right side of the equation, we have (x + 2)^2 = 15/2, which is in vertex form.

6. To solve for x, we take the square root of both sides, giving us x + 2 = ±sqrt(15/2), or x = -2 ±sqrt(15/2).

Thus, completing the square enabled us to rewrite a quadratic equation in vertex form and solve for x at the same time.

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