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Simplifying 5 + 6y(y + 2) = 40 Reorder the terms: 5 + 6y(2 + y) = 40 5 + (2 * 6y + y * 6y) = 40 5 + (12y + 6y2) = 40 Solving 5 + 12y + 6y2 = 40 Solving for variable 'y'. Reorder the terms: 5 + -40 + 12y + 6y2 = 40 + -40 Combine like terms: 5 + -40 = -35 -35 + 12y + 6y2 = 40 + -40 Combine like terms: 40 + -40 = 0 -35 + 12y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -5.833333333 + 2y + y2 = 0 Move the constant term to the right: Add '5.833333333' to each side of the equation. -5.833333333 + 2y + 5.833333333 + y2 = 0 + 5.833333333 Reorder the terms: -5.833333333 + 5.833333333 + 2y + y2 = 0 + 5.833333333 Combine like terms: -5.833333333 + 5.833333333 = 0.000000000 0.000000000 + 2y + y2 = 0 + 5.833333333 2y + y2 = 0 + 5.833333333 Combine like terms: 0 + 5.833333333 = 5.833333333 2y + y2 = 5.833333333 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 5.833333333 + 1 Reorder the terms: 1 + 2y + y2 = 5.833333333 + 1 Combine like terms: 5.833333333 + 1 = 6.833333333 1 + 2y + y2 = 6.833333333 Factor a perfect square on the left side: (y + 1)(y + 1) = 6.833333333 Calculate the square root of the right side: 2.614064523 Break this problem into two subproblems by setting (y + 1) equal to 2.614064523 and -2.614064523.Subproblem 1
y + 1 = 2.614064523 Simplifying y + 1 = 2.614064523 Reorder the terms: 1 + y = 2.614064523 Solving 1 + y = 2.614064523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 2.614064523 + -1 Combine like terms: 1 + -1 = 0 0 + y = 2.614064523 + -1 y = 2.614064523 + -1 Combine like terms: 2.614064523 + -1 = 1.614064523 y = 1.614064523 Simplifying y = 1.614064523Subproblem 2
y + 1 = -2.614064523 Simplifying y + 1 = -2.614064523 Reorder the terms: 1 + y = -2.614064523 Solving 1 + y = -2.614064523 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -2.614064523 + -1 Combine like terms: 1 + -1 = 0 0 + y = -2.614064523 + -1 y = -2.614064523 + -1 Combine like terms: -2.614064523 + -1 = -3.614064523 y = -3.614064523 Simplifying y = -3.614064523Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.614064523, -3.614064523}
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