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Simplifying 2(2.5 + 0.75y) * 3y = 4 Reorder the terms for easier multiplication: 2 * 3y(2.5 + 0.75y) = 4 Multiply 2 * 3 6y(2.5 + 0.75y) = 4 (2.5 * 6y + 0.75y * 6y) = 4 (15y + 4.5y2) = 4 Solving 15y + 4.5y2 = 4 Solving for variable 'y'. Reorder the terms: -4 + 15y + 4.5y2 = 4 + -4 Combine like terms: 4 + -4 = 0 -4 + 15y + 4.5y2 = 0 Begin completing the square. Divide all terms by 4.5 the coefficient of the squared term: Divide each side by '4.5'. -0.8888888889 + 3.333333333y + y2 = 0 Move the constant term to the right: Add '0.8888888889' to each side of the equation. -0.8888888889 + 3.333333333y + 0.8888888889 + y2 = 0 + 0.8888888889 Reorder the terms: -0.8888888889 + 0.8888888889 + 3.333333333y + y2 = 0 + 0.8888888889 Combine like terms: -0.8888888889 + 0.8888888889 = 0.0000000000 0.0000000000 + 3.333333333y + y2 = 0 + 0.8888888889 3.333333333y + y2 = 0 + 0.8888888889 Combine like terms: 0 + 0.8888888889 = 0.8888888889 3.333333333y + y2 = 0.8888888889 The y term is 3.333333333y. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333y + 2.777777779 + y2 = 0.8888888889 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333y + y2 = 0.8888888889 + 2.777777779 Combine like terms: 0.8888888889 + 2.777777779 = 3.6666666679 2.777777779 + 3.333333333y + y2 = 3.6666666679 Factor a perfect square on the left side: (y + 1.666666667)(y + 1.666666667) = 3.6666666679 Calculate the square root of the right side: 1.914854216 Break this problem into two subproblems by setting (y + 1.666666667) equal to 1.914854216 and -1.914854216.Subproblem 1
y + 1.666666667 = 1.914854216 Simplifying y + 1.666666667 = 1.914854216 Reorder the terms: 1.666666667 + y = 1.914854216 Solving 1.666666667 + y = 1.914854216 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = 1.914854216 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = 1.914854216 + -1.666666667 y = 1.914854216 + -1.666666667 Combine like terms: 1.914854216 + -1.666666667 = 0.248187549 y = 0.248187549 Simplifying y = 0.248187549Subproblem 2
y + 1.666666667 = -1.914854216 Simplifying y + 1.666666667 = -1.914854216 Reorder the terms: 1.666666667 + y = -1.914854216 Solving 1.666666667 + y = -1.914854216 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = -1.914854216 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = -1.914854216 + -1.666666667 y = -1.914854216 + -1.666666667 Combine like terms: -1.914854216 + -1.666666667 = -3.581520883 y = -3.581520883 Simplifying y = -3.581520883Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.248187549, -3.581520883}
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