Additional solved examples for splitting the middle term

Example 1: x² - 12x + 35

Solution:

Get the master product and middle term:	Master product = first term * constant = 35x2 Middle term = -12x
Split the middle term into two parts whose product equals the master product:	-12x = -5x - 7x (& -5x * -7x = 35x2 = master product)
Rewrite the expression with the middle term split into two parts:	x2 - 5x - 7x + 35
Make two groups, with each group consisting of two terms:	(x2 - 5x) + (-7x + 35)
Factor out the common factors from each group:	x(x - 5) - 7(x - 5)
Take (x - 5) as the common factor:	(x - 5)(x - 7)

Answer: (x - 5)(x - 7)

Example 2: 8x² - 26x + 15

Solution:

Get the master product and middle term:	Master product = first term * constant number = 8x2 * 15 = 120x2 Middle term = -26x
Split the middle term into two parts whose product equals the master product:	-26x = -20x - 6x (& -20x * -6x = 120x2)
Rewrite the quadratic expression with the two parts:	8x2 - 20x - 6x + 15
Group the four termed polynomial so formed into two groups, with each group containing two terms:	(8x2 - 20x) + (-6x + 15)
Factor out common factors from each group:	4x(2x - 5) - 3(2x - 5)
Take (2x - 5) as the common factor:	(2x - 5)(4x - 3)

Answer: (2x - 5)(4x - 3)

Example 3: 2x² + 5x - 25

Solution:

Calculate the master product and middle term:	Master product = first term * constant = 2x2 * -25 = -50x2 Middle term = 5x
Divide the middle term 5x into two parts such that their product equals the master product -50x2	5x = 10x - 5x (& 10x * -5x = -50x2)
Rewrite the quadratic expression with the two parts of the middle term:	2x2 + 10x - 5x - 25
Group the four terms into two groups with each group containing two terms:	(2x2 + 10x) + (-5x - 25)
Factor out common factors from each group:	2x(x + 5) - 5(x + 5)
T(x + 5) is the common factor:	(x + 5)(2x - 5)

Answer: (x + 5)(2x - 5)

Example 4: √3x² + 11x + 6√3

Solution:

Calculate the master product and middle term of the given quadratic expression:	Master product = first term * constant = √3x2 * 6√3 = 18x2 Middle term = 11x
Split the middle term into two parts such that their product equals the master product:	11x = 9x + 2x (& 9x * 2x = 18x2 = master product of quadratic)
Rewrite the quadratic with the two parts of the middle term:	√3x2 + 9x + 2x + 6√3
Group into two groups with each group containing two terms:	(√3x2 + 9x) + (2x + 6√3)
Take common factors from each group:	√3x(x + 3√3) + 2(x + 3√3)
Take (x + 3√3) as a common factor:	(x + 3√3)(√3x + 3)

Answer: (x + 3√3)(√3x + 2)

Example 5: 3a²x² + 8abx + 4b²

Solution:

Calculate the master product and middle term:	Master product = first term * constant(or third term) = 3a2x2 * 4b2 = 12a2x2b2 Middle term: 8abx
Split the middle term into two parts whose product equals the master product:	8abx = 6abx + 2abx (& 6abx * 2abx = 12a2x2b2 = master product)
Rewrite the quadratic expression with the two parts of the middle term:	3a2x2 + 6abx + 2abx + 4b2
Group the four terms into two groups with each group containing two terms:	(3a2x2 + 6abx) + (2abx + 4b2)
Take common factors from each group:	3ax(ax + 2b) + 2b(ax + 2b)
Take (ax + 2b) as the common factor:	(ax + 2b)(3ax + 2b)

Answer: (ax + 2b)(3ax + 2b)