## Completing the Square

Completing the Square is an algebraic method used to solve quadratic equations by transforming them into a perfect square trinomial. It is essential in understanding the relationship between rational and irrational numbers, as the process often reveals the nature of a quadratic equation’s roots. This technique involves manipulating integers, square and square roots to rewrite the equation in a simpler form, which aids in finding exact solutions. In statistics, it’s useful for least squares methods, providing insights into the numerical behavior of data. The method also helps in visualizing quadratic functions, revealing their minimum or maximum values.

## What is Completing the Square?

Completing the Square is a method used to rewrite a quadratic equation of the form ๐๐ฅยฒ+๐๐ฅ+๐ = 0 into a perfect square form (๐ฅโ๐)ยฒ = ๐. This transformation makes it easier to solve for the roots of the equation by isolating ๐ฅ. The process involves adding and subtracting a specific value derived from the coefficient of ๐ฅ to create a perfect square trinomial. Once in this form, the equation can be solved by taking the square root on both sides and then isolating ๐ฅ.

## Completing the Square Method

Completing the Square is often used to factorize quadratic equations and identify their roots or zeros, especially when traditional factorization methods fail. A quadratic equation like ๐๐ฅยฒ+๐๐ฅ+๐=0 can sometimes be challenging to factor due to its complexity. In such cases, completing the square offers a viable solution.

### Consider an example to understand this better:

If you have a quadratic equation that can’t be easily factorized, completing the square can rearrange it into a more manageable form. This approach involves rewriting ๐๐ฅยฒ+๐๐ฅ+๐ as a perfect square trinomial, making it easier to solve for ๐ฅ.

## Completing the Square Steps

**Standard Form:**Start with a quadratic equation in standard form: ๐๐ฅยฒ+๐๐ฅ+๐ = 0.**Isolate ๐ฅยฒ and ๐ฅ:**If ๐ (the coefficient of ๐ฅยฒ) is not 1, divide the entire equation by ๐*a*to make the coefficient of ๐ฅยฒ equal to 1.**Move ๐**Move the constant term ๐*c*:*c*to the right side of the equation.**Find the Completing Term:**Take half of the coefficient of ๐ฅ, ๐/2โ, and then square it to get the completing term: (๐/2)ยฒ.**Add and Subtract the Completing Term:**Add this completing term to both sides of the equation to keep the equation balanced.**Form a Perfect Square:**The left side of the equation can now be factored into a perfect square trinomial, (๐ฅ+๐/2)ยฒ, which is equal to the right side.**Solve for ๐ฅ:**Take the square root of both sides and solve for ๐ฅ. Remember to consider both the positive and negative square roots.**Simplify the Solution:**Isolate ๐ฅ to find the solution(s).

## How to Apply Completing the Square Method?

Here’s an example with a different quadratic equation:

### Example: Complete the square for 3๐ฅยฒ+12๐ฅ+9

**Step 1:**

Factor out the coefficient of ๐ฅ2*x*2, which is 33:

3๐ฅยฒ+12๐ฅ+9 = 3(๐ฅยฒ+4๐ฅ+3)

Now, the coefficient of ๐ฅยฒ is 1.

**Step 2:**

Find half of the coefficient of ๐ฅ*x* in the factored expression.

Here, the coefficient of ๐ฅ is . Half of 4 is 2.

**Step 3:**

Square this number to find (4/2)ยฒ = 4.

**Step 4:**

Add and subtract 4 after the ๐ฅ term inside the parentheses:

3(๐ฅยฒ+4๐ฅ+4โ4+3)

**Step 5:**

Factorize the perfect square trinomial formed by the first three terms:

๐ฅยฒ+4๐ฅ+4 = (๐ฅ+2)ยฒ

So the expression becomes:

3((๐ฅ+2)ยฒโ4+3)

**Step 6:**

Simplify the constant terms:

โ4+3 = โ1

So, the final expression is:

3((๐ฅ+2)ยฒโ1) = 3(๐ฅ+2)ยฒโ3

This completes the square for the expression 3๐ฅยฒ+12๐ฅ+9 as:

3(๐ฅ+2)ยฒโ3

To summarize:

- Ensure the coefficient of ๐ฅยฒ is 1.
- Add and subtract (๐/2๐)ยฒ inside the parentheses.
- Factorize the perfect square trinomial.
- Simplify the expression to obtain the result.

## Completing the Square Formula

The formula for completing the square simplifies the process of converting a quadratic equation into its vertex form. For a quadratic expression ๐๐ฅยฒ+๐๐ฅ+๐, we can rewrite it using the formula ๐(๐ฅ+๐)ยฒ+๐, where ๐ and ๐*n* are calculated as ๐ = ๐/2๐โ and ๐ = ๐โ๐ยฒ/4๐โ. This approach allows us to quickly identify the vertex of the parabola represented by the quadratic equation. By substituting the values of ๐*m* and ๐*n* back into the formula, we transform the equation into a format that highlights the parabola’s vertex. For example, given 2๐ฅยฒ+8๐ฅ+5, we find ๐ = 2 and ๐ = โ3, which results in the equivalent form 2(๐ฅ+2)ยฒโ3. This method provides a streamlined way to manipulate quadratic expressions and understand their geometric properties.

## Completing the Square Formula Examples

### Example 1:

Given 3๐ฅยฒ+12๐ฅ+7

**Calculate ๐:**

๐ = 12/2ร3 = 12/6=2**Calculate ๐:**

๐ = 7โ12ยฒ/4ร3 = 7โ144/12 = 7โ12 = โ5**Substitute into the Formula:**

Substitute ๐ and ๐ into the formula:

3๐ฅยฒ+12๐ฅ+7 = 3(๐ฅ+2)ยฒโ5

### Example 2:

Given โ2๐ฅยฒ+4๐ฅโ1

**Calculate ๐:**

๐ = 4/2รโ2 = 4/โ4 = โ1**Calculate ๐**

๐ = โ1โ4ยฒ/4รโ2 = โ1โ16/โ8 = โ1+2 = 1**Substitute into the Formula:**

Substitute ๐*m*and ๐ into the formula:

โ2๐ฅยฒ+4๐ฅโ1 = โ2(๐ฅโ1)ยฒ+1

## Derivation of Completing the Square Formula

To complete the square in the expression ๐๐ฅยฒ+๐๐ฅ+๐*a*, we follow these steps:

**Normalize the Coefficient of ๐ฅ2:**

Factor out ๐*a*to make the coefficient of ๐ฅยฒ equal to 1:

๐๐ฅยฒ+๐๐ฅ+๐ = ๐[๐ฅยฒ+๐๐๐ฅ+๐๐]**Analyze the First Two Terms:**

Consider the terms ๐ฅยฒ and (๐/๐)๐ฅ. The term ๐ฅยฒ represents the area of a square with side length ๐ฅ. The term (๐/๐)๐ฅ represents the area of a rectangle with length ๐/๐โ and breadth ๐ฅ.**Visualize the Problem Geometrically:**

Draw a square with side length ๐ฅ, giving it an area of ๐ฅยฒ. Then, add a rectangle next to it with length ๐๐*ab*โ and breadth ๐ฅ, forming a new composite shape.**Adjust the Rectangle into Two Identical Parts:**

The rectangle’s length ๐/๐โ can be split into two halves of ๐/2๐โ each. Attach these halves on either side of the original square to create a new shape.**Create a New Larger Square:**

To complete the larger square, we need to fill in the missing square in the corner. The missing square’s side length is ๐/2๐โ, and its area is (๐/2๐)ยฒ = ๐ยฒ/4๐ยฒโ.**Add and Subtract the Missing Area:**

Add and subtract the missing area inside the factored expression: ๐[๐ฅ2+(๐/๐)๐ฅ+๐ยฒ/4๐ยฒโ๐ยฒ/4๐ยฒ+๐/๐]โ**Factor the Perfect Square Trinomial:**

The first three terms form a perfect square trinomial: ๐[(๐ฅ+๐/2๐)ยฒโ๐ยฒ/4๐ยฒ+๐/๐]**Simplify the Expression:**

Simplify the expression further: ๐๐ฅยฒ+๐๐ฅ+๐ = ๐(๐ฅ+๐/2๐)ยฒ+๐(๐/๐โ๐ยฒ/4๐ยฒ)

To complete the square for a quadratic expression like ๐๐ฅยฒ+๐๐ฅ+๐, the first step is to factor out ๐*a* to normalize the coefficient of ๐ฅยฒ to 1. This transforms the equation into ๐[๐ฅยฒ+๐๐๐ฅ+๐๐]. Focusing on the first two terms, ๐ฅยฒ and (๐/๐)๐ฅ, we then calculate half the coefficient of ๐ฅ, which is ๐/2๐โ. Squaring this value yields ๐ยฒ/4๐ยฒโ. We add and subtract this square inside the expression, creating a perfect square trinomial. The equation now looks like ๐ฅยฒ+(๐/๐)๐ฅ = (๐ฅ+๐/2๐)ยฒโ๐ยฒ4๐ยฒโ. By substituting this back into the original equation, we obtain ๐๐ฅยฒ+๐๐ฅ+๐ = ๐[(๐ฅ+๐/2๐)ยฒโ๐ยฒ4๐ยฒ]+๐. Expanding this and simplifying gives the quadratic equation in completed square form: ๐(๐ฅ+๐/2๐)ยฒ+(๐โ๐ยฒ/4๐). This transformation provides an easy way to see the vertex of the parabola described by the quadratic equation and simplifies further calculations.

## Trick to Learn Completing the Square Method

**Normalize the Quadratic Coefficient:**

If the coefficient of ๐ฅยฒ isn’t 1, factor it out of the first two terms to ensure the quadratic coefficient is 1.**Add and Subtract the Missing Square:**

Take half the coefficient of ๐ฅ, square it, and add and subtract this value after the ๐ฅ term. This completes the square inside the expression.**Simplify into Completed Square Form:**

Rewrite the expression in the form (๐ฅ+๐)ยฒ by factoring the trinomial and adjust the constants outside to find the final expression in the form ๐(๐ฅ+๐)ยฒ+๐.

## FAQs

## What is the purpose of the completing the square method?

The purpose of completing the square is to rewrite a quadratic expression in a different form, specifically ๐(๐ฅ+๐)ยฒ+๐. This transformation makes it easier to solve quadratic equations and identify the vertex of a parabola.

## How do you handle a quadratic equation if the coefficient of ๐ฅยฒis not 1?

If the coefficient of ๐ฅยฒ isn’t 1, factor out that coefficient from the quadratic and linear terms to normalize it to 1. For example, with 4๐ฅยฒ+8๐ฅ, factor out 4 to get 4(๐ฅยฒ+2๐ฅ).

## What does adding and subtracting the square achieve in the completing the square method?

Adding and subtracting the square helps create a perfect square trinomial. This trinomial can then be rewritten as a squared binomial, which is essential to putting the quadratic equation into the desired completed square form.

## How does completing the square relate to the quadratic formula?

Completing the square is a method that can be used to derive the quadratic formula. It transforms the quadratic equation into a form where taking the square root directly reveals the values of ๐ฅ.