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Simplifying (5 + 3y) * y = 11.7 Reorder the terms for easier multiplication: y(5 + 3y) = 11.7 (5 * y + 3y * y) = 11.7 (5y + 3y2) = 11.7 Solving 5y + 3y2 = 11.7 Solving for variable 'y'. Reorder the terms: -11.7 + 5y + 3y2 = 11.7 + -11.7 Combine like terms: 11.7 + -11.7 = 0.0 -11.7 + 5y + 3y2 = 0.0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -3.9 + 1.666666667y + y2 = 0.0 Move the constant term to the right: Add '3.9' to each side of the equation. -3.9 + 1.666666667y + 3.9 + y2 = 0.0 + 3.9 Reorder the terms: -3.9 + 3.9 + 1.666666667y + y2 = 0.0 + 3.9 Combine like terms: -3.9 + 3.9 = 0.0 0.0 + 1.666666667y + y2 = 0.0 + 3.9 1.666666667y + y2 = 0.0 + 3.9 Combine like terms: 0.0 + 3.9 = 3.9 1.666666667y + y2 = 3.9 The y term is 1.666666667y. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667y + 0.6944444447 + y2 = 3.9 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667y + y2 = 3.9 + 0.6944444447 Combine like terms: 3.9 + 0.6944444447 = 4.5944444447 0.6944444447 + 1.666666667y + y2 = 4.5944444447 Factor a perfect square on the left side: (y + 0.8333333335)(y + 0.8333333335) = 4.5944444447 Calculate the square root of the right side: 2.143465522 Break this problem into two subproblems by setting (y + 0.8333333335) equal to 2.143465522 and -2.143465522.Subproblem 1
y + 0.8333333335 = 2.143465522 Simplifying y + 0.8333333335 = 2.143465522 Reorder the terms: 0.8333333335 + y = 2.143465522 Solving 0.8333333335 + y = 2.143465522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = 2.143465522 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = 2.143465522 + -0.8333333335 y = 2.143465522 + -0.8333333335 Combine like terms: 2.143465522 + -0.8333333335 = 1.3101321885 y = 1.3101321885 Simplifying y = 1.3101321885Subproblem 2
y + 0.8333333335 = -2.143465522 Simplifying y + 0.8333333335 = -2.143465522 Reorder the terms: 0.8333333335 + y = -2.143465522 Solving 0.8333333335 + y = -2.143465522 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + y = -2.143465522 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = -2.143465522 + -0.8333333335 y = -2.143465522 + -0.8333333335 Combine like terms: -2.143465522 + -0.8333333335 = -2.9767988555 y = -2.9767988555 Simplifying y = -2.9767988555Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.3101321885, -2.9767988555}
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