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Simplifying 2y(4y + 10) = 5y + 32 Reorder the terms: 2y(10 + 4y) = 5y + 32 (10 * 2y + 4y * 2y) = 5y + 32 (20y + 8y2) = 5y + 32 Reorder the terms: 20y + 8y2 = 32 + 5y Solving 20y + 8y2 = 32 + 5y Solving for variable 'y'. Reorder the terms: -32 + 20y + -5y + 8y2 = 32 + 5y + -32 + -5y Combine like terms: 20y + -5y = 15y -32 + 15y + 8y2 = 32 + 5y + -32 + -5y Reorder the terms: -32 + 15y + 8y2 = 32 + -32 + 5y + -5y Combine like terms: 32 + -32 = 0 -32 + 15y + 8y2 = 0 + 5y + -5y -32 + 15y + 8y2 = 5y + -5y Combine like terms: 5y + -5y = 0 -32 + 15y + 8y2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4 + 1.875y + y2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 1.875y + 4 + y2 = 0 + 4 Reorder the terms: -4 + 4 + 1.875y + y2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 1.875y + y2 = 0 + 4 1.875y + y2 = 0 + 4 Combine like terms: 0 + 4 = 4 1.875y + y2 = 4 The y term is 1.875y. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875y + 0.87890625 + y2 = 4 + 0.87890625 Reorder the terms: 0.87890625 + 1.875y + y2 = 4 + 0.87890625 Combine like terms: 4 + 0.87890625 = 4.87890625 0.87890625 + 1.875y + y2 = 4.87890625 Factor a perfect square on the left side: (y + 0.9375)(y + 0.9375) = 4.87890625 Calculate the square root of the right side: 2.208824631 Break this problem into two subproblems by setting (y + 0.9375) equal to 2.208824631 and -2.208824631.Subproblem 1
y + 0.9375 = 2.208824631 Simplifying y + 0.9375 = 2.208824631 Reorder the terms: 0.9375 + y = 2.208824631 Solving 0.9375 + y = 2.208824631 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + y = 2.208824631 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + y = 2.208824631 + -0.9375 y = 2.208824631 + -0.9375 Combine like terms: 2.208824631 + -0.9375 = 1.271324631 y = 1.271324631 Simplifying y = 1.271324631Subproblem 2
y + 0.9375 = -2.208824631 Simplifying y + 0.9375 = -2.208824631 Reorder the terms: 0.9375 + y = -2.208824631 Solving 0.9375 + y = -2.208824631 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + y = -2.208824631 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + y = -2.208824631 + -0.9375 y = -2.208824631 + -0.9375 Combine like terms: -2.208824631 + -0.9375 = -3.146324631 y = -3.146324631 Simplifying y = -3.146324631Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.271324631, -3.146324631}
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