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Solve the equation: 8y^3 - 216 = 0. Show your work.

a) y = 6
b) y = 3
c) y = -3
d) y = -6

User Arafat
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1 Answer

5 votes

Final answer:

To solve the equation 8y^3 - 216 = 0, factor out the greatest common factor, recognize 27 as a perfect cube, set each factor equal to zero, and solve using the quadratic formula. The solution is y = 3, corresponding to option b.

Step-by-step explanation:

To solve the equation 8y^3 - 216 = 0, we can start by factoring out the greatest common factor, which is 8:

8(y^3 - 27) = 0

Next, we recognize that 27 is a perfect cube, so we can rewrite it as (3^3):

8(y - 3)(y^2 + 3y + 9) = 0

Setting each factor equal to zero, we have:

y - 3 = 0 --> y = 3

y^2 + 3y + 9 = 0

Using the quadratic formula, we find two complex solutions for y^2 + 3y + 9 = 0:

y = (-3 ± √(-3^2 - 4(1)(9)))/(2(1))

y = (-3 ± √(-3^2 - 36))/2

y = (-3 ± √(-45))/2

y = (-3 ± 3√5*i)/2

Therefore, the solutions to the equation are y = 3 or y = (-3 ± 3√5*i)/2. Since we are only considering real solutions, the correct answer is y = 3, which corresponds to option b.

User Squeazer
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