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Simplifying (10y2) + (y2 + -6) + (y + 1) = 180 Reorder the terms: (10y2) + (-6 + y2) + (y + 1) = 180 Remove parenthesis around (-6 + y2) (10y2) + -6 + y2 + (y + 1) = 180 Reorder the terms: (10y2) + -6 + y2 + (1 + y) = 180 Remove parenthesis around (1 + y) (10y2) + -6 + y2 + 1 + y = 180 Reorder the terms: -6 + 1 + y + (10y2) + y2 = 180 Combine like terms: -6 + 1 = -5 -5 + y + (10y2) + y2 = 180 Combine like terms: (10y2) + y2 = 11y2 -5 + y + 11y2 = 180 Solving -5 + y + 11y2 = 180 Solving for variable 'y'. Reorder the terms: -5 + -180 + y + 11y2 = 180 + -180 Combine like terms: -5 + -180 = -185 -185 + y + 11y2 = 180 + -180 Combine like terms: 180 + -180 = 0 -185 + y + 11y2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -16.81818182 + 0.09090909091y + y2 = 0 Move the constant term to the right: Add '16.81818182' to each side of the equation. -16.81818182 + 0.09090909091y + 16.81818182 + y2 = 0 + 16.81818182 Reorder the terms: -16.81818182 + 16.81818182 + 0.09090909091y + y2 = 0 + 16.81818182 Combine like terms: -16.81818182 + 16.81818182 = 0.00000000 0.00000000 + 0.09090909091y + y2 = 0 + 16.81818182 0.09090909091y + y2 = 0 + 16.81818182 Combine like terms: 0 + 16.81818182 = 16.81818182 0.09090909091y + y2 = 16.81818182 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.09090909091y + 0.25 + y2 = 16.81818182 + 0.25 Reorder the terms: 0.25 + 0.09090909091y + y2 = 16.81818182 + 0.25 Combine like terms: 16.81818182 + 0.25 = 17.06818182 0.25 + 0.09090909091y + y2 = 17.06818182 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 17.06818182 Calculate the square root of the right side: 4.131365612 Break this problem into two subproblems by setting (y + 0.5) equal to 4.131365612 and -4.131365612.Subproblem 1
y + 0.5 = 4.131365612 Simplifying y + 0.5 = 4.131365612 Reorder the terms: 0.5 + y = 4.131365612 Solving 0.5 + y = 4.131365612 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 4.131365612 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 4.131365612 + -0.5 y = 4.131365612 + -0.5 Combine like terms: 4.131365612 + -0.5 = 3.631365612 y = 3.631365612 Simplifying y = 3.631365612Subproblem 2
y + 0.5 = -4.131365612 Simplifying y + 0.5 = -4.131365612 Reorder the terms: 0.5 + y = -4.131365612 Solving 0.5 + y = -4.131365612 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -4.131365612 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -4.131365612 + -0.5 y = -4.131365612 + -0.5 Combine like terms: -4.131365612 + -0.5 = -4.631365612 y = -4.631365612 Simplifying y = -4.631365612Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.631365612, -4.631365612}
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