Last updated on July 30th, 2024 at 11:20 pm

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Completing the Square is a method in mathematics that is used for converting a quadratic expression of the form ( ax^{2} + bx + c ) to the vertex form ( a(x + m)^{2} + n ). The most common use of this method is in solving a quadratic equation, which can be done by rearranging the expression obtained after completing the square.

## Completing the Square: Method, Formula, Examples

Completing the Square is a powerful algebraic technique used to solve quadratic equations, rewrite quadratic expressions, and find the vertex of a parabola. This blog post will guide you through the method, and formula, and provide examples to help you understand how to apply this technique effectively.

What is Completing the Square?

Completing the square involves transforming a quadratic expression of the form ( ax^{2} + bx + c ) into a perfect square trinomial plus a constant. The goal is to rewrite the quadratic expression in the vertex form ( a(x + m)^{2} + n ).

## Method of Completing the Square

### Step-by-Step Guide

1. Start with the standard quadratic form:

\[ ax^2 + bx + c \]

2. Divide all terms by \( a \) (if \( a \neq 1 \)):

\[ x^2 + \frac{b}{a}x + \frac{c}{a} \]

3. Isolate the constant term on one side:

\[ x^2 + \frac{b}{a}x = -\frac{c}{a} \]

4. Find the value that completes the square:

Take half of the coefficient of \( x \), square it, and add it to both sides.

\[ \left( \frac{\frac{b}{a}}{2} \right)^2 = \left( \frac{b}{2a} \right)^2 \]

5. Add this value to both sides:

\[ x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 \]

6. Rewrite the left side as a square:

\[ \left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 \]

7. Simplify the right side:

\[ \left( x + \frac{b}{2a} \right)^2 = \frac{b^2 – 4ac}{4a^2} \]

8. Rewrite the equation in vertex form:

\[ a(x + \frac{b}{2a})^2 + \text{constant} \]

#### Example 1: Solving a Quadratic Equation

Solve the quadratic equation \( x^2 + 6x + 5 = 0 \) by completing the square.

1. Start with the equation:

\[ x^2 + 6x + 5 = 0 \]

2. Move the constant term to the other side:

\[ x^2 + 6x = -5 \]

3. Complete the square:

\[ \left( \frac{6}{2} \right)^2 = 9 \]

\[ x^2 + 6x + 9 = -5 + 9 \]

\[ (x + 3)^2 = 4 \]

4. Solve for \( x \):

\[ x + 3 = \pm 2 \]

\[ x = -3 \pm 2 \]

\[ x = -1 \text{ or } x = -5 \]

#### Example 2: Rewriting a Quadratic Expression

Rewrite the quadratic expression \( 2x^2 + 8x + 6 \) by completing the square.

1. Start with the expression:

\[ 2x^2 + 8x + 6 \]

2. Factor out the coefficient of \( x^2 \):

\[ 2(x^2 + 4x) + 6 \]

3. Complete the square inside the parentheses:

\[ \left( \frac{4}{2} \right)^2 = 4 \]

\[ 2(x^2 + 4x + 4 – 4) + 6 \]

\[ 2((x + 2)^2 – 4) + 6 \]

4. Distribute and simplify:

\[ 2(x + 2)^2 – 8 + 6 \]

\[ 2(x + 2)^2 – 2 \]

#### Example 3: Finding the Vertex of a Parabola

Find the vertex of the quadratic function \( f(x) = 3x^2 – 12x + 7 \) by completing the square.

1. Start with the function:

\[ f(x) = 3x^2 – 12x + 7 \]

2. Factor out the coefficient of \( x^2 \):

\[ f(x) = 3(x^2 – 4x) + 7 \]

3. Complete the square inside the parentheses:

\[ \left( \frac{-4}{2} \right)^2 = 4 \]

\[ f(x) = 3(x^2 – 4x + 4 – 4) + 7 \]

\[ f(x) = 3((x – 2)^2 – 4) + 7 \]

4. Distribute and simplify:

\[ f(x) = 3(x – 2)^2 – 12 + 7 \]

\[ f(x) = 3(x – 2)^2 – 5 \]

5. Identify the vertex:

\[ \text{Vertex: } (2, -5) \]

#### Example 4: Solving a Quadratic Equation with Non-Unit Leading Coefficient

Solve the quadratic equation \( 2x^2 + 4x – 6 = 0 \) by completing the square.

1. Start with the equation:

\[ 2x^2 + 4x – 6 = 0 \]

2. Divide all terms by 2:

\[ x^2 + 2x – 3 = 0 \]

3. Move the constant term to the other side:

\[ x^2 + 2x = 3 \]

4. Complete the square:

\[ \left( \frac{2}{2} \right)^2 = 1 \]

\[ x^2 + 2x + 1 = 3 + 1 \]

\[ (x + 1)^2 = 4 \]

5. Solve for \( x \):

\[ x + 1 = \pm 2 \]

\[ x = -1 \pm 2 \]

\[ x = 1 \text{ or } x = -3 \]

#### Example 5: Rewriting a Quadratic Expression with Fractional Coefficients

Rewrite the quadratic expression \( \frac{1}{2}x^2 + x + \frac{3}{2} \) by completing the square.

1. Start with the expression:

\[ \frac{1}{2}x^2 + x + \frac{3}{2} \]

2. Factor out the coefficient of \( x^2 \):

\[ \frac{1}{2}(x^2 + 2x) + \frac{3}{2} \]

3. Complete the square inside the parentheses:

\[ \left( \frac{2}{2} \right)^2 = 1 \]

\[ \frac{1}{2}(x^2 + 2x + 1 – 1) + \frac{3}{2} \]

\[ \frac{1}{2}((x + 1)^2 – 1) + \frac{3}{2} \]

4. Distribute and simplify:

\[ \frac{1}{2}(x + 1)^2 – \frac{1}{2} + \frac{3}{2} \]

\[ \frac{1}{2}(x + 1)^2 + 1 \]

#### Example 6: Solving a Quadratic Equation with a Complex Solution

Solve the quadratic equation \( x^2 + 4x + 8 = 0 \) by completing the square.

1. Start with the equation:

\[ x^2 + 4x + 8 = 0 \]

2. Move the constant term to the other side:

\[ x^2 + 4x = -8 \]

3. Complete the square:

\[ \left( \frac{4}{2} \right)^2 = 4 \]

\[ x^2 + 4x + 4 = -8 + 4 \]

\[ (x + 2)^2 = -4 \]

4. Solve for \( x \):

\[ x + 2 = \pm \sqrt{-4} \]

\[ x + 2 = \pm 2i \]

\[ x = -2 \pm 2i \]

Exercises

1. Solve the quadratic equation \( x^2 + 10x + 16 = 0 \) by completing the square.