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:
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 quadraticequation 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.
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
- their product is -3x2, and
- their sum is 2x
Step 4
Write the highest common factors (along with the signs) of each row to its left
and of each column on its top.
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).
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
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