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.