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Generic Rectangle Calculator

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


If you are new to factoring quadratics using generic rectangles and are looking for a generic rectangle calculator to make your job easier, you have come to the right place. Whether you have a math assignment or are just interested in expanding your knowledge of quadratic equations, we have all the information you need. Keep reading to learn:

  • What quadratic equations are;
  • What generic rectangles are;
  • What the term factoring means;
  • How do generic rectangles help us to solve quadratic equations; and
  • How to use our generic rectangle solver.

If you wish to explore other methods used to solve quadratic equations, be sure to check our completing the square calculator, the quadratic formula calculator, and the factoring trinomials calculator.

What is a quadratic equation?

A quadratic equation is an algebraic equation of a second degree. It is usually written as:

ax2 + bx + c = 0

In this equation, x is an unknown value while a, b, and c are constants. In the quadratic equation, a cannot be equal to 0. The expression at the left-hand side of the equal sign is called a quadratic trinomial.

A quadratic equation can have zero, one, or two solutions (in real numbers). When there is only one solution, it is called a double root.

What are generic rectangles?

Generic rectangles are rectangles used in mathematics to multiply large numbers. The word generic refers to the fact that we do not draw these rectangles to scale.

We use generic rectangles in this calculator to find the factors of quadratic equations.

🙋 Do you remember what a factor is in mathematics? When we multiply two numbers together to form another number, the first two numbers are called factors of the product.

So, factoring means finding the numbers that can divide a given number without leaving a remainder.

Our generic rectangle solver helps you to find two expressions, which, when multiplied together, will result in the given quadratic trinomial. Then we can immediately answer the question when our trinomial assumes the value zero, which is equivalent to solving a quadratic equation.

How to factor quadratic equations using a generic rectangle

For our example, we will use the trinomial 4x2+12x+5=04x^2+12x+5 = 0.

Now before we begin, be sure to draw your generic rectangle like the one shown in the diagram below.

A rectangle with no set dimensions.

If you examine any quadratic trinomial, you will notice that each has a term containing x2x^2 and a constant term (it may be equal to zero!).

Let's begin:

  1. First, we place the term containing the x2x^2, in our case 4x24x^2, and the constant terms, in again 55, at opposite ends of the generic rectangle diagonally, like this:
A generic rectangle with the x^2 and constant terms at diagonally opposite ends of the rectangle.
  1. Next, find the product of 4x24x^2 and of the constant term 55. The answer in this example is 20x220x^2.
  1. List all the factors of the coefficient of the product found above.

    Since this coefficient is 2020 the factors are:

    2020 and 11
    1010 and 22
    55 and 44

  1. Identify the two factors which, when added together, will result in the coefficient of the x term, i.e., in 1212.

    As you can see by now, these are 1010 and 22.

  1. Multiply these numbers by xx and place them in the empty places in the rectangle:
Generic rectangle showing the products of the smaller parts.
  1. Our next task is to find the common factors of each of the smaller sections of our rectangle: of each row and each column. As you can see:

    • 5×2x=10x5 × 2x = 10x, so we write 2x2x and 55 along the length and width of the rectangle containing the 10x10x. If you study the diagram below, you will see that the others follow the same pattern.

    • 2x×2x=4x22x × 2x = 4x^2

    • 5×1=55 × 1 = 5 and

    • 1×2x=2x1 × 2x = 2x.

    We write them outside the rectangle. These are the factors of our equation.

Generic rectangle showing factors.
  1. By looking at what now represents the length and width of the rectangle, we can rewrite our answers as follows:

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

Hence, the equation 4x2+12x+5=04x^2 +12x+5 = 0 takes on the form (2x+5)(2x+1)=0(2x + 5)(2x+1) = 0, which we can solve very easily.

Namely, we must have 2x+5=02x + 5 = 0 or 2x+1=02x+1 = 0. This means that x=52x = -\frac 52 or x=12x = -\frac 12. And that is how we solve quadratic equations using generic rectangles!

Do these steps seem too long and complicated for you? Just use our generic rectangle calculator, and you will be done in a jiffy 😁.

How to use our general rectangle calculator?

Using our generic rectangle solver is quite simple, really. Enter the three constant terms, a, b, c, and our calculator will return the factors and the generic rectangle. It can also provide an explanation of how to factor the equation.

To display a detailed explanation of the steps used to derive the answer, change the default no to yes at the bottom of the calculator.

FAQ

What are the factors of the quadratic equation with coefficients 1, 3, and 2 respectively?

(x+2)(x+1).

We factor this equation using generic rectangles. You can verify this answer by expanding the parentheses in the usual manner.

You may also use Omni calculator's generic rectangle calculator to solve the problem quickly.

How to factor a quadratic equation using a generic rectangle?

To find the factors:

  1. Place the term containing x2 at the bottom left of the rectangle and the constant term at the top right corner.

  2. Compute their product and list its factors.

  3. Find the factors of the x2 term, which, when added together, result in the x term. Put them in empty cells.

  4. Find the common factors of each row and each column in the rectangle.

  5. The length times the width of the rectangle is the factorization of your equation.

Adena Benn
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This calculator uses the generic rectangle method to factor the quadratic equation:

ax² + bx + c

with integer coefficients a, b, c.
Enter the coefficients:
a
b
c
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