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To factorize the quadratic expression 6v² - 27v + 27, you can follow these steps:

Step 1: Look for common factors among the coefficients of the quadratic expression. In this case, there are no common factors among 6, -27, and 27.

Step 2: Multiply the coefficient of the quadratic term (6) by the constant term (27). The result is 162.

Step 3: Find two numbers that multiply to give the result from step 2 (162) and add up to the coefficient of the linear term (-27). In this case, the numbers are -9 and -18 because (-9) * (-18) = 162 and (-9) + (-18) = -27.

Step 4: Rewrite the middle term (-27v) using the two numbers found in step 3. Split the middle term as -9v - 18v.

Step 5: Group the terms and factor by grouping. Rearrange the expression by grouping the first two terms and the last two terms:

(6v² - 9v) + (-18v + 27)

Step 6: Factor out the greatest common factor from each group:

3v(2v - 3) - 9(2v - 3)

Step 7: Notice that the terms in the parentheses are the same. Factor out the common binomial (2v - 3):

(2v - 3)(3v - 9)

Therefore, the quadratic expression 6v² - 27v + 27 can be factorized as (2v - 3)(3v - 9).

Step 1: Look for common factors among the coefficients of the quadratic expression. In this case, there are no common factors among 6, -27, and 27.

Step 2: Multiply the coefficient of the quadratic term (6) by the constant term (27). The result is 162.

Step 3: Find two numbers that multiply to give the result from step 2 (162) and add up to the coefficient of the linear term (-27). In this case, the numbers are -9 and -18 because (-9) * (-18) = 162 and (-9) + (-18) = -27.

Step 4: Rewrite the middle term (-27v) using the two numbers found in step 3. Split the middle term as -9v - 18v.

Step 5: Group the terms and factor by grouping. Rearrange the expression by grouping the first two terms and the last two terms:

(6v² - 9v) + (-18v + 27)

Step 6: Factor out the greatest common factor from each group:

3v(2v - 3) - 9(2v - 3)

Step 7: Notice that the terms in the parentheses are the same. Factor out the common binomial (2v - 3):

(2v - 3)(3v - 9)

Therefore, the quadratic expression 6v² - 27v + 27 can be factorized as (2v - 3)(3v - 9).

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