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### 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:
Step 2
• In box 1, write the first term x2
• In box 4, write the third term -3
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.
Step 4
Write the highest common factors (along with the signs) of each row to its left
and of each column on its top.
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).
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