Box Method of factoring quadractic expressions

The Box Method is an easy to understand, visual method of factoring a quadratic expression into a product of two linear expressions. It is best learnt with the help of an example.

x² + 3x – 4

Factor the above quadratic by the Box Method.

Solution:

1. Draw a box and divide it into four equal
parts:

2. Write the first term of the quadratic (whose coefficient is x²) in box 1 and the third term (ie
the constant) in box 4.

3.

The product of box 1 and box 4 is –4x2. Find two algebraic terms which when multiplied give the result –4x2 and that add up to the middle term of the quadratic, 3x. Henceforth write these terms in boxes 2 and 3. Here the two algebraic terms are –x and 4x, because –x + 4x equals 3x and –x times 4x equals –4x2   Then, write these two terms in boxes 1 and 3.

4. Write the highest common factor of boxes 1 and 2 on the left of box 1:

5. Similarly, write the highest common factors of the lower row to its left, and those of each column on its top:

6. Form an algebraic expression by taking the two terms on the left side of the boxes and adding them. Form another algebraic expression by taking the two terms on top of the boxes and adding them. These two algebraic expressions are the factors of the given quadratic expression, and their product is equal to the quadratic expression. Thus, we have factored the quadratic into a product of two linear expressions.


Taking two terms from the left, we obtain

(x + 4)

Taking two terms from the top,we obtain

(x – 1)

Therefore,    x² + 3x – 4 = (x + 4)(x – 1)