0,5у^3-0, 5у(у+1) (у-3) =7

To solve this equation, let's simplify it step by step.

Step 1: Distribute the -0.5y to the terms inside the parentheses:

0.5y^3 - 0.5y(у+1)(у-3) = 7

0.5y^3 - 0.5y(у^2 - 2у - 3) = 7

Step 2: Distribute -0.5y to each term inside the parentheses:

0.5y^3 - 0.5y^3 + y^2 - 1.5y - 1.5y + 4.5 = 7

Step 3: Combine like terms:

y^2 - 3y + 4.5 = 7

Step 4: Subtract 7 from both sides:

y^2 - 3y + 4.5 - 7 = 0

y^2 - 3y - 2.5 = 0

Now we have a quadratic equation in terms of y. To solve for y, you can either use factoring, completing the square, or the quadratic formula. Let's solve it using the quadratic formula.

Step 1: Identify the values of a, b, and c in the quadratic equation (ax^2 + bx + c = 0).

In our equation, a = 1, b = -3, and c = -2.5.

Step 2: Substitute the values into the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

y = (-(-3) ± √((-3)^2 - 4(1)(-2.5))) / (2(1))

Simplify:

y = (3 ± √(9 + 10)) / 2

y = (3 ± √(19)) / 2

Therefore, the solutions for y are:

y = (3 + √19) / 2

y = (3 - √19) / 2