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### Box method additional solved examples

Example 1: x2 - 8x + 15 = 0

Solution:

Step 1:  Draw a box, divide it into
four equal boxes. Write the first term of the quadratic in box 1 and third term
in box 4.

Step 2:  Find two algebraic terms whose
product equals the product of box 1 and box 4, and whose sum equals the middle term
(or second term) of the quadratic. Write these two terms in boxes 2 and 3 respectively.

Step 3:  Calculate the highest common
factor of each row and column and write it on the left of the row or at the top
of the column.

Step 4:  The quadratic expression has
been factored. Take the two terms on the left of the rows and put a plus sign between
them. Similarly take the two terms at the top of the two columns and put a plus
sign between them. The product of these two algebraic expressions is equal to the

x2 - 8x + 15 = (x - 3)(x - 5)

Step 5:  Calculate the roots by applying
the Zero Product Rule

The quadratic equation obtained is: (x - 3)(x - 5) = 0. By applying the Zero Product
Rule, either (x - 3) is equal to zero, or (x - 5) is equal to zero.

Since (x - 3)(x - 5) = 0,

• Either (x - 3) = 0, whence x = 3
• Or (x - 5) = 0, whence x = 5

Example 2: 16x = 4x2 + 15

Solution:

First convert the given quadratic equation to the standard form:

4x2 - 16x + 15 = 0
Step 1:  Draw a box, divide it into
four equal boxes. Write the first term of the quadratic in box 1 and third term
in box 4.

Step 2:  Find two algebraic terms whose
product equals the product of box 1 and box 4, and whose sum equals the middle term
(or second term) of the quadratic. Write these two terms in boxes 2 and 3 respectively.

Step 3:  Calculate the highest common
factor of each row and column and write it on the left of the row or at the top
of the column.

Step 4:  The quadratic expression has
been factored. Take the two terms on the left of the rows and put a plus sign between
them. Similarly take the two terms at the top of the two columns and put a plus
sign between them. The product of these two algebraic expressions is equal to the

4x2 - 16x + 15 = (2x - 3)(2x - 5)

Step 5:  Calculate the roots by applying
the Zero Product Rule

The quadratic equation obtained is: (2x - 3)(2x - 5) = 0. By applying the Zero Product
Rule, either (2x - 3) is equal to zero, or (2x - 5) is equal to zero.

Since (2x - 3)(2x - 5) = 0, by applying the Zero Product Rule,

• Either (2x - 3) = 0, whence x = 3/2
• Or (2x - 5) = 0, whence x = 5/2

Example 3: 3x2 + 5 = 8x

Solution:

Convert the given quadratic equation to the standard form,

3x2 - 8x + 5 = 0

Step 1:  Draw a box, divide it into
four equal boxes. Write the first term of the quadratic in box 1 and third term
in box 4.

Step 2:  Find two algebraic terms whose
product equals the product of box 1 and box 4, and whose sum equals the middle term
(or second term) of the quadratic. Write these two terms in boxes 2 and 3 respectively.

Step 3:  Calculate the highest common
factor of each row and column and write it on the left of the row or at the top
of the column.

Step 4:  The quadratic expression has
been factored. Take the two terms on the left of the rows and put a plus sign between
them. Similarly take the two terms at the top of the two columns and put a plus
sign between them. The product of these two algebraic expressions is equal to the

3x2 - 8x + 5 = (x - 1)(3x - 5)

Step 5:  Calculate the roots by applying
the Zero Product Rule

The quadratic equation obtained is: (x - 1)(3x - 5) = 0. By applying the Zero Product
Rule, either (x - 1) is equal to zero, or (3x - 5) is equal to zero.

Since (x - 1)(3x - 5) = 0, by applying Zero Product Rule,

• Either x - 1 = 0, whence x = 1
• Or 3x - 5 = 0, whence x = 5/3

Example 4: 2x2 + 5x = 12

Solution:

Convert the given quadratic equation to the standard form,

2x2 + 5x - 12 = 0
Step 1:  Draw a box, divide it into
four equal boxes. Write the first term of the quadratic in box 1 and third term
in box 4.

Step 2:  Find two algebraic terms whose
product equals the product of box 1 and box 4, and whose sum equals the middle term
(or second term) of the quadratic. Write these two terms in boxes 2 and 3 respectively.

Step 3:  Calculate the highest common
factor of each row and column and write it on the left of the row or at the top
of the column.

Step 4:  The quadratic expression has
been factored. Take the two terms on the left of the rows and put a plus sign between
them. Similarly take the two terms at the top of the two columns and put a plus
sign between them. The product of these two algebraic expressions is equal to the

2x2 + 5x - 12 = (x + 4)(2x - 3)

Step 5:  Calculate the roots by applying
the Zero Product Rule

The quadratic equation obtained is: (x + 4)(2x - 3) = 0. By applying the Zero Product
Rule, either (x + 4) is equal to zero, or (2x - 3) is equal to zero.

Since (x + 4)(2x - 3) = 0, by applying the Zero Product Rule,

• Either (x + 4) = 0, whence x = -4
• Or (2x - 3) = 0, whence x = 3/2