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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)