Completing the Square for a Binomial

Oct 13, 2025 by leolavanderia.com.br 34 views

Introduction

In algebra, completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This method is particularly useful in solving quadratic equations and in simplifying expressions. In this article, we will focus on completing the square for a binomial and then factoring the resulting perfect square trinomial.

What is a Binomial?

A binomial is a polynomial expression with two terms. It can be written in the form of $ax + b$ , where $a$ and $b$ are constants, and $x$ is the variable. For example, $x + 3$ and $2x - 4$ are binomials.

Completing the Square for a Binomial

To complete the square for a binomial, we need to add a constant term to the expression so that it becomes a perfect square trinomial. The constant term to be added is the square of half the coefficient of the variable term.

Example 1: Completing the Square for $x^2 - 12x$

Let's consider the binomial $x^2 - 12x$ . To complete the square, we need to add a constant term to this expression.

Step 1: Identify the Coefficient of the Variable Term

The coefficient of the variable term $x$ is $-12$ .

Step 2: Calculate the Constant Term to be Added

To calculate the constant term to be added, we need to square half the coefficient of the variable term. Half of $-12$ is $-6$ , and the square of $-6$ is $36$ .

Step 3: Add the Constant Term to the Expression

Now, we add $36$ to the expression $x^2 - 12x$ to form a perfect square trinomial.

$x^2 - 12x + 36$

The resulting expression is a perfect square trinomial.

Example 2: Completing the Square for $x^2 + 8x$

Let's consider the binomial $x^2 + 8x$ . To complete the square, we need to add a constant term to this expression.

Step 1: Identify the Coefficient of the Variable Term

The coefficient of the variable term $x$ is $8$ .

Step 2: Calculate the Constant Term to be Added

To calculate the constant term to be added, we need to square half the coefficient of the variable term. Half of $8$ is $4$ , and the square of $4$ is $16$ .

Step 3: Add the Constant Term to the Expression

Now, we add $16$ to the expression $x^2 + 8x$ to form a perfect square trinomial.

$x^2 + 8x + 16$

The resulting expression is a perfect square trinomial.

Factoring the Resulting Perfect Square Trinomial

A perfect square trinomial can be factored into the square of a binomial. The general form of a perfect square trinomial is $(x + a)^2$ , where $a$ is a constant.

Example 1: Factoring the Perfect Square Trinomial $x^2 - 12x + 36$

Let's consider the perfect square trinomial $x^2 - 12x + 36$ . To factor this expression, we need to find the binomial whose square is equal to the given expression.

Step 1: Identify the Binomial Whose Square is Equal to the Given Expression

The binomial whose square is equal to $x^2 - 12x + 36$ is $x - 6$ .

Step 2: Write the Perfect Square Trinomial as the Square of the Binomial

Now, we can write the perfect square trinomial $x^2 - 12x + 36$ as the square of the binomial $x - 6$ .

$(x - 6)^2$

The resulting expression is the square of the binomial $x - 6$ .

Example 2: Factoring the Perfect Square Trinomial $x^2 + 8x + 16$

Let's consider the perfect square trinomial $x^2 + 8x + 16$ . To factor this expression, we need to find the binomial whose square is equal to the given expression.

Step 1: Identify the Binomial Whose Square is Equal to the Given Expression

The binomial whose square is equal to $x^2 + 8x + 16$ is $x + 4$ .

Step 2: Write the Perfect Square Trinomial as the Square of the Binomial

Now, we can write the perfect square trinomial $x^2 + 8x + 16$ as the square of the binomial $x + 4$ .

$(x + 4)^2$

The resulting expression is the square of the binomial $x + 4$ .

Conclusion

Introduction

In our previous article, we discussed how to complete the square for a binomial and factor the resulting perfect square trinomial. In this article, we will answer some frequently asked questions related to completing the square and factoring perfect square trinomials.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to convert a quadratic expression into a perfect square trinomial. This method is particularly useful in solving quadratic equations and in simplifying expressions.

Q: How do I know if a quadratic expression can be completed to a perfect square trinomial?

A: A quadratic expression can be completed to a perfect square trinomial if it has the form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a \neq 0$ . If the expression has a constant term, it can be completed to a perfect square trinomial.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

$x^2 + bx + \left(\frac{b}{2}\right)^2$

where $b$ is the coefficient of the variable term.

Q: How do I factor a perfect square trinomial?

A: A perfect square trinomial can be factored into the square of a binomial. The general form of a perfect square trinomial is $(x + a)^2$ , where $a$ is a constant.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

Not adding the correct constant term
Not squaring the correct value
Not factoring the resulting perfect square trinomial correctly

Q: Can I complete the square for a quadratic expression with a negative leading coefficient?

A: Yes, you can complete the square for a quadratic expression with a negative leading coefficient. However, you need to be careful when adding the constant term, as it may change the sign of the expression.

Q: How do I know if a perfect square trinomial can be factored into the square of a binomial?

A: A perfect square trinomial can be factored into the square of a binomial if it has the form $(x + a)^2$ , where $a$ is a constant.

Q: What are some real-world applications of completing the square and factoring perfect square trinomials?

A: Some real-world applications of completing the square and factoring perfect square trinomials include:

Solving quadratic equations in physics and engineering
Simplifying expressions in algebra and calculus
Factoring polynomials in number theory and cryptography

Conclusion

In this article, we have answered some frequently asked questions related to completing the square and factoring perfect square trinomials. We hope that this article has provided you with a better understanding of these concepts and has helped you to overcome any challenges you may have faced.