Box method of solving quadratic equations

The box method is a visual way of factoring quadratic expressions. It changes a
quadratic expression from a trinomial to a product of two linear expressions. Thus
the box method factors a quadratic expression.

A quadratic equation must be in the standard form in order to solve it by the box
method. The LHS of a quadratic equation in the standard form is a quadratic expression
of the form of ax2 + bx + c. This expression is factored to the form
p(x - q)(x - r) in the box method. The roots are calculated by applying the zero
product rule on the equation thus obtained.

How to do the box method:

This section describes how the box method can solve quadratic equations. The quadratic
equation x2 + 2x - 3 = 0 is solved by the box method. The working is
divided into six steps:



Step 1
Draw a box and divide it into four equal smaller boxes:
Box method: Step 1 (image)
Step 2
  • In box 1, write the first term x2
  • In box 4, write the third term -3
Box method: Step 2 (image)
Step 3
Box 2 and box 3 are to be filled with terms such that the product of box 2 and box
3 equals the product of box 1 and box 4 and the sum of box 2 and box 3 equals the
middle (second) term of the quadratic, 2x.

  • Product of box 1 and box 4 = -3x2
  • Middle (or second) term = 2x
Therefore the two required two terms are 3x and -x, because,


  • their product is -3x2, and
  • their sum is 2x
Thus write the two terms 3x and -x in boxes 2 and 3 respectively.
Box method: Step 3 (image)
Step 4
Write the highest common factors (along with the signs) of each row to its left
and of each column on its top.
Box method: Step 4 (image)
Step 5
Take the two terms on the left of the box and make an algebraic expression by putting
a plus sign between them, that is, (x - 1).


Similarly, take the two terms on the top of the box and make an algebraic expression
by putting a plus sign between them, that is, (x + 3)


The product of these two algebraic expressions, (x - 1) and (x + 3) is equal to
the original quadratic expression. Thus the quadratic expression x2 +
2x - 3 has been factored to (x - 1)(x + 3).
Box method: Step 5 (image)
Step 6
Applying the Zero Product Rule to the quadratic equation (x - 1)(x + 3) = 0,


  • Either (x - 1) = 0, whence x = 1
  • Or (x + 3) = 0, whence x = -3
Box method: Step 6 (image)

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