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To solve the quadratic equation 6v² - 27v + 27, you can use the quadratic formula or factorization method.

1. Quadratic Formula:

The quadratic formula is given by:

v = (-b ± √(b² - 4ac)) / (2a)

For the equation 6v² - 27v + 27, the coefficients are:

a = 6, b = -27, c = 27

Substituting these values into the quadratic formula, we get:

v = (-(-27) ± √((-27)² - 4(6)(27))) / (2(6))

v = (27 ± √(729 - 648)) / 12

v = (27 ± √81) / 12

Simplifying further:

v = (27 ± 9) / 12

This gives two possible solutions:

v₁ = (27 + 9) / 12 = 36 / 12 = 3

v₂ = (27 - 9) / 12 = 18 / 12 = 1.5

Therefore, the solutions to the equation 6v² - 27v + 27 are v = 3 and v = 1.5.

2. Factorization Method:

To factorize the quadratic equation 6v² - 27v + 27, we can look for two numbers whose product is equal to ac (6 * 27 = 162) and whose sum is equal to b (-27).

The factors of 162 are: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162

After trying different combinations, we find that 9 and 18 satisfy the conditions:

6v² - 27v + 27 = 6v² - 9v - 18v + 27

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

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

Setting each factor equal to zero:

3v - 9 = 0 or 2v - 3 = 0

Solving these equations gives:

3v = 9 or 2v = 3

v = 3 or v = 3/2 = 1.5

Therefore, the solutions to the equation 6v² - 27v + 27 are v = 3 and v = 1.5, which matches the solutions obtained using the quadratic formula.

1. Quadratic Formula:

The quadratic formula is given by:

v = (-b ± √(b² - 4ac)) / (2a)

For the equation 6v² - 27v + 27, the coefficients are:

a = 6, b = -27, c = 27

Substituting these values into the quadratic formula, we get:

v = (-(-27) ± √((-27)² - 4(6)(27))) / (2(6))

v = (27 ± √(729 - 648)) / 12

v = (27 ± √81) / 12

Simplifying further:

v = (27 ± 9) / 12

This gives two possible solutions:

v₁ = (27 + 9) / 12 = 36 / 12 = 3

v₂ = (27 - 9) / 12 = 18 / 12 = 1.5

Therefore, the solutions to the equation 6v² - 27v + 27 are v = 3 and v = 1.5.

2. Factorization Method:

To factorize the quadratic equation 6v² - 27v + 27, we can look for two numbers whose product is equal to ac (6 * 27 = 162) and whose sum is equal to b (-27).

The factors of 162 are: 1, 2, 3, 6, 9, 18, 27, 54, 81, 162

After trying different combinations, we find that 9 and 18 satisfy the conditions:

6v² - 27v + 27 = 6v² - 9v - 18v + 27

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

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

Setting each factor equal to zero:

3v - 9 = 0 or 2v - 3 = 0

Solving these equations gives:

3v = 9 or 2v = 3

v = 3 or v = 3/2 = 1.5

Therefore, the solutions to the equation 6v² - 27v + 27 are v = 3 and v = 1.5, which matches the solutions obtained using the quadratic formula.

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