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Box Method Calculator

Created by Adena Benn
Reviewed by Anna Szczepanek, PhD and Steven Wooding
Last updated: Jan 18, 2024


The box method calculator is just what you need to relieve the stress some of us experience at the thought of algebra. This calculator is a powerful tool that allows you to enter the coefficients of your trinomial expressions, and it returns the factors as well as the details of the actual calculation used to solve the expression.

This accompanying article also contains valuable information on the following:

  • What the box method is in maths;
  • Step-by-step explanation on how to do the box method factoring by hand;
  • How to use the box method factoring calculator; and
  • The difference between polynomials and trinomials.

Are you interested in learning more about factoring trinomials? Visit our completing the square calculator, the factoring trinomials calculator or this quadratic formula calculator.

What is the box method in mathematics?

In math, the box method is a procedure used to factor trinomials. It utilizes a rectangle or box that is not drawn to scale. The box is divided into four parts, and then we find and allocate values to sections of the box, for which we later find the highest common factors.

What is a trinomial?

A trinomial is an algebraic expression (a polynomial) with three terms that are either added or subtracted. Trinomials consist of a mixture of variables and constants. A trinomial is usually written as ax² + bx + c, where a, b, and c are numbers.

How to do the box method calculation by hand

Let's suppose that you have been given the following trinomial to factorize:

2x²+7x+5=02x² + 7x + 5 = 0

The coefficients and the constant term are referred to as aa, bb, and cc, respectively. That is:

ax²+bx+c=0ax² + bx + c = 0

Here are the steps:

  1. We first need to draw the box and divide it into four parts.

 

 

  1. Next, place the aa and cc terms in the box diagonally opposite each other.

 

55

2x22x^2

  1. Find the product of the coefficients of the values you just placed in the box. In our case, it's 1010.

  2. Now, find the factors of the product found in Step 3.

    11 and 1010
    22 and 55

  3. Find the two factors which, when added together, will result in the coefficient of the middle term of the original equation.

    22 and 55

  4. Now attach xx to these numbers and place the 2x2x and 5x5x terms in the empty places in the box.

5x5x

55

2x22x^2

2x2x

  1. Find the highest common factor of each pair of terms in each row and each column:

    • In the first row, we get: 55;

    • In the second row: 2x2x;

    • In the first column: xx; and

    • In the second column: 11.

  2. Take the sum of the terms produced by the rows:

    2x+52x + 5

    This is our first factor!

  3. Take the sum of the terms produced by the columns:

    x+1x + 1

    This is the second factor!

  4. So, the factors of the trinomial 2x²+7x+5=02x² + 7x + 5 = 0 are:

    (2x+5)(x+1)(2x + 5)(x + 1)

If you are interested in other algebraic calculators, visit the adding and subtracting polynomials calculator or our absolute value inequalities calculator.

How to use our box method factoring calculator

Our box method calculator is a straightforward tool that requires you to enter the coefficients of your equation, and it will return the correct factors of the expression.

As a bonus, you may also view the box method calculation with steps by changing the option for Show step-by-step solution.

FAQ

How do I factor trinomials using the box method?

Here is how we use the box method to factor trinomials:

  1. Write the first and last terms in the box diagonally opposite each other.

  2. Find the factors of the product of the coefficients of the first and last terms.

  3. Identify the factors of the product which, when added together, will result in the middle term.

  4. Attach x to these terms and place them in the empty places in the box.

  5. Place the factors of each row and column at the bottom and left of the box.

  6. Add the terms on each side of the box.

What are the factors of the trinomial 3x² - 4x - 4?

(x - 2)(3x + 2). We factor this equation using the box method. You could also do this faster using Omni's box method factoring calculator.

What is the difference between polynomials and trinomials?

In mathematics, a polynomial is an expression that contains several terms that are products of variables and coefficients. A trinomial is a polynomial containing exactly three terms. In particular, a quadratic trinomial contains an x^2 term, an x term, and a constant term.

Adena Benn
Coefficients
This calculator uses the box method to factor the quadratic equation:

ax² + bx + c

with integer coefficients a, b, c.
Enter the coefficients:
a
b
c
Show step-by-step solution?
No, thanks.
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