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Simplifying 186 = (5y)(4y + 3) Remove parenthesis around (5y) 186 = 5y(4y + 3) Reorder the terms: 186 = 5y(3 + 4y) 186 = (3 * 5y + 4y * 5y) 186 = (15y + 20y2) Solving 186 = 15y + 20y2 Solving for variable 'y'. Reorder the terms: 186 + -15y + -20y2 = 15y + -15y + 20y2 + -20y2 Combine like terms: 15y + -15y = 0 186 + -15y + -20y2 = 0 + 20y2 + -20y2 186 + -15y + -20y2 = 20y2 + -20y2 Combine like terms: 20y2 + -20y2 = 0 186 + -15y + -20y2 = 0 Begin completing the square. Divide all terms by -20 the coefficient of the squared term: Divide each side by '-20'. -9.3 + 0.75y + y2 = 0 Move the constant term to the right: Add '9.3' to each side of the equation. -9.3 + 0.75y + 9.3 + y2 = 0 + 9.3 Reorder the terms: -9.3 + 9.3 + 0.75y + y2 = 0 + 9.3 Combine like terms: -9.3 + 9.3 = 0.0 0.0 + 0.75y + y2 = 0 + 9.3 0.75y + y2 = 0 + 9.3 Combine like terms: 0 + 9.3 = 9.3 0.75y + y2 = 9.3 The y term is 0.75y. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75y + 0.140625 + y2 = 9.3 + 0.140625 Reorder the terms: 0.140625 + 0.75y + y2 = 9.3 + 0.140625 Combine like terms: 9.3 + 0.140625 = 9.440625 0.140625 + 0.75y + y2 = 9.440625 Factor a perfect square on the left side: (y + 0.375)(y + 0.375) = 9.440625 Calculate the square root of the right side: 3.072560008 Break this problem into two subproblems by setting (y + 0.375) equal to 3.072560008 and -3.072560008.Subproblem 1
y + 0.375 = 3.072560008 Simplifying y + 0.375 = 3.072560008 Reorder the terms: 0.375 + y = 3.072560008 Solving 0.375 + y = 3.072560008 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + y = 3.072560008 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + y = 3.072560008 + -0.375 y = 3.072560008 + -0.375 Combine like terms: 3.072560008 + -0.375 = 2.697560008 y = 2.697560008 Simplifying y = 2.697560008Subproblem 2
y + 0.375 = -3.072560008 Simplifying y + 0.375 = -3.072560008 Reorder the terms: 0.375 + y = -3.072560008 Solving 0.375 + y = -3.072560008 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + y = -3.072560008 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + y = -3.072560008 + -0.375 y = -3.072560008 + -0.375 Combine like terms: -3.072560008 + -0.375 = -3.447560008 y = -3.447560008 Simplifying y = -3.447560008Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.697560008, -3.447560008}
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