Example 1: x

Solution:

^{2}- 8x + 15 = 0Solution:

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

original quadratic expression.

x

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,

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

original quadratic expression.

x

^{2}- 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 = 4x

Solution:

^{2}+ 15Solution:

First convert the given quadratic equation to the standard form:

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

original quadratic expression.

4x

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,

4x

Step 1: Draw a box, divide it into^{2}- 16x + 15 = 0four 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

original quadratic expression.

4x

^{2}- 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: 3x

Solution:

^{2}+ 5 = 8xSolution:

Convert the given quadratic equation to the standard form,

3x

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

original quadratic expression.

3x

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,

3x

^{2}- 8x + 5 = 0Step 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

original quadratic expression.

3x

^{2}- 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: 2x

Solution:

^{2}+ 5x = 12Solution:

Convert the given quadratic equation to the standard form,

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

original quadratic expression.

2x

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,

2x

Step 1: Draw a box, divide it into^{2}+ 5x - 12 = 0four 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

original quadratic expression.

2x

^{2}+ 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

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