Table of Contents

**Completing the Square Formula**

**Introduction to Completing the Square Formula**

Completing the square is a method that is used for converting a quadratic expression of the form ax2 + bx + c to the vertex form a(x – h)2 + k. The most common application of completing the square is in solving a quadratic equation. This can be done by rearranging the expression obtained after completing the square: a(x + m)2 + n, such that the left side is a perfect square trinomial.

Let us learn more about completing the square formula, its method and the process of completing the square step-wise. We will discuss its applications using solved examples for a better understanding.

**What is Completing the Square?**

Completing the square is a method in algebra that is used to write a quadratic expression in a way such that it contains the perfect square. In simple words, we can say that completing the square is a process where consider a quadratic equation of the ax2 + bx + c = 0 and change it to write it in perfecting the square form a(x + p)2 + q = 0.

Completing the square method is useful in:

- Converting a quadratic expression from standard form into vertex form.

- Analyzing at which point the quadratic expression has minimum/maximum value (vertex).
- Graphing a quadratic function.
- Solving a quadratic equation.
- Deriving the quadratic formula.

**Completing the Square Method**

The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots or zeros of a quadratic polynomial or a quadratic equation. We know that a quadratic equation of the form ax2 + bx + c = 0 can be solved by the factorization method. But sometimes, factorizing the quadratic expression ax2 + bx + c is complex or NOT possible. Let us have a look at the following example to understand this case.

For example:

x2 + 2x + 3 cannot be factorized as we cannot find two numbers whose sum is 2 and whose product is 3. In such cases, we write it in the form a(x + m)2 + n by completing the square. Since we have (x + m) whole squared, we say that we have “completed the square” here. But, how do we complete the square? Let us understand the concept in detail in the following sections.

**Completing the Square Steps**

To apply the method of completing the square, we will follow a certain set of steps. Given below is the process of completing the square stepwise:

- Step 1: Write the quadratic equation as x2 + bx + c. (Coefficient of x2 needs to be 1. If not, take it as the common factor.)
- Step 2: Determine half of the coefficient of x.
- Step 3: Take the square of the number obtained in step 1.

- Step 4: Add and subtract the square obtained in step 2 to the x2 term.
- Step 5: Factorize the polynomial and apply the algebraic identity x2 + 2xy + y2 = (x + y)2 (or) x2 – 2xy + y2 = (x – y)2 to complete the square.

**Completing the Square Formula**

Completing the square formula is a technique or method to convert a quadratic polynomial or equation into a perfect square with some additional constant.

A quadratic expression in variable x: ax2 + bx + c, where a, b and c are any real numbers but a ≠ 0, can be converted into a perfect square with some additional constant by using completing the square formula or technique.

Note: Completing the square formula is used to derive the quadratic formula.

Completing the square formula is a technique or method that can also be used to find the roots of the given quadratic equations, ax2 + bx + c = 0, where a, b and c are any real numbers but a ≠ 0.

**Formula for Completing the Square**

The formula for completing the square is: ax2 + bx + c ⇒ a(x + m)2 + n, where

- m = b/2a and
- n = c – (b2/4a)

Instead of using the complex step-wise method for completing the square, we can use the following simple formula to complete the square. To complete the square in the expression ax2 + bx + c, first find the values of m and n using the above formulas and then substitute these values in: ax2 + bx + c = a(x + m)2 + n. These formulas are derived geometrically.

**Trick to Learn Completing the Square Method**

Here are a few tips for completing the square formula.

- Step 1: Note down the form we wish to obtain after completing the square: a(x + m)2 + n

- Step 2: After expanding, we get, ax2 + 2amx + am2 + n
- Step 3: Compare the given expression, say ax2 + bx + c and find m and n as m = b/2a and n = c – (b2/4a).

**Solved Examples on Completing the Square Examples**

**Example 1:** Use completing the square method to solve: x2 – 4x – 5 = 0.

Solution:

Let us transpose the constant term to the other side of the equation:

x2 – 4x = 5

Now, take half of the coefficient of the x-term, which is -4, including the sign, which gives -2. Take the square of -2 to get +4, and add this squared value to both sides of the equation:

x2 – 4x + 4 = 5 + 4

⇒ x2 – 4x + 4 = 9

By using one of the algebraic identities, we can write x2 – 4x + 4 = (x – 2)2.

(x – 2)2 = 9

Now that we have completed the expression to create a perfect-square binomial, let us solve:

(x – 2)2 = 9

⇒ x – 2 = ±3

⇒ x = 2 ± 3

⇒ x = 5, -1

**Example 2:** Complete the square in the quadratic expression 2x2 + 7x + 6.

Solution:

The given expression is 2x2 + 7x + 6. The first step to complete the square is to make the coefficient of x2 as 1. We will take the coefficient of x2 (which is 2) as a common factor.

2x2 + 7x + 6 = 2(x2 + (7/2)x + 3) → Equation (1)

The coefficient of x is 7/2. Half of it is 7/4. Its square is (7/4)2 = 49/16.

[This term can also be found using (b/2a)2 = [7/2(2)]2 = 49/16]

Add and subtract it after the x term in Equation (1):

2x2 + 7x + 6 = 2(x2 + (7/2)x + 49/4 – 49/4 + 3)

Factorize the trinomial made by the first three terms:

2x2 + 7x + 6 = 2(x2 + (7/2)x + (49/16) – (49/16) + 3)

= 2[(x2 + (7/4))2 – (49/16) + 3]

= 2((x2 + (7/4))2 – (1/16))

= 2(x2 + (7/4))2 – 1/8)

The final answer is of the form a(x + m)2 + n and hence perfecting the square has been done.

**Frequently Asked Questions on Completing the Square Examples**

### What is the Method of Completing the Square?

Completing the square is a method in mathematics that is used for converting a quadratic expression of the form ax2 + 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.

### What is the Easiest Way to Learn to complete the Square?

The easiest way to learn to complete the square method is using the formula, a(x + m)2 + n = a(x + m)2 + n. Here, m and n can be calculated with the help of the following formulas, m = b/2a and, n = c - (b2/4a).

### What is the Use of Completing the Square?

Completing the square formula is used for the following purposes:

- Converting a quadratic expression into vertex form.
- Computing the vertex of a quadratic function.
- Graphing a quadratic function.
- Finding the roots of a quadratic equation.

### What to Add When Completing the Square?

If we have the expression ax2 + bx + c, then we need to add and subtract (b/2a)2 which will complete the square in the expression. This will result in {x + (b/a)}2 - (b/2a)2 + c.

### How do you Complete the Square With two Variables?

Consider an expression in two variables x2 + y2 + 2x + 4y + 7. To complete the square, we take each of the coefficients of x and y, make their value half, and then square it.

The coefficient of x = 2, the coefficient of y = 4.

This means, (1/2 × 2)2 = 1 and (1/2 × 4)2 = 4.

Let us add and subtract this to the given equation. Then, rearrange the terms to complete the squares.

x2 + y2 + 2x + 4y + 7 + (1 - 1) + (4 - 4) = (x2 + 2x + 1) + (y2 + 4y + 4) + 7 - 1 - 4 = (x + 1)2 + (y + 2)2 + 2

### When to use Perfecting the Square?

Answer: We use the perfecting the square method when we want to convert a quadratic expression of the form ax2 + bx + c to the vertex form a(x - h)2 + k.

### What is Completing the Square Formula?

Completing the square formula is the formula required to convert a quadratic polynomial or equation into a perfect square with some additional constant. It is expressed as, ax2 + bx + c ⇒ a(x + m)2 + n, where, m and n are real numbers.

### What is the Use of Completing the Square Formula?

Completing the square formula is used when we want to represent a quadratic polynomial or equation into a perfect square with some additional constant and thus used to factorize a quadratic polynomial.