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Simplifying 42y + 6 = 14 + 2y + -25y2 Reorder the terms: 6 + 42y = 14 + 2y + -25y2 Solving 6 + 42y = 14 + 2y + -25y2 Solving for variable 'y'. Reorder the terms: 6 + -14 + 42y + -2y + 25y2 = 14 + 2y + -25y2 + -14 + -2y + 25y2 Combine like terms: 6 + -14 = -8 -8 + 42y + -2y + 25y2 = 14 + 2y + -25y2 + -14 + -2y + 25y2 Combine like terms: 42y + -2y = 40y -8 + 40y + 25y2 = 14 + 2y + -25y2 + -14 + -2y + 25y2 Reorder the terms: -8 + 40y + 25y2 = 14 + -14 + 2y + -2y + -25y2 + 25y2 Combine like terms: 14 + -14 = 0 -8 + 40y + 25y2 = 0 + 2y + -2y + -25y2 + 25y2 -8 + 40y + 25y2 = 2y + -2y + -25y2 + 25y2 Combine like terms: 2y + -2y = 0 -8 + 40y + 25y2 = 0 + -25y2 + 25y2 -8 + 40y + 25y2 = -25y2 + 25y2 Combine like terms: -25y2 + 25y2 = 0 -8 + 40y + 25y2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.32 + 1.6y + y2 = 0 Move the constant term to the right: Add '0.32' to each side of the equation. -0.32 + 1.6y + 0.32 + y2 = 0 + 0.32 Reorder the terms: -0.32 + 0.32 + 1.6y + y2 = 0 + 0.32 Combine like terms: -0.32 + 0.32 = 0.00 0.00 + 1.6y + y2 = 0 + 0.32 1.6y + y2 = 0 + 0.32 Combine like terms: 0 + 0.32 = 0.32 1.6y + y2 = 0.32 The y term is 1.6y. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6y + 0.64 + y2 = 0.32 + 0.64 Reorder the terms: 0.64 + 1.6y + y2 = 0.32 + 0.64 Combine like terms: 0.32 + 0.64 = 0.96 0.64 + 1.6y + y2 = 0.96 Factor a perfect square on the left side: (y + 0.8)(y + 0.8) = 0.96 Calculate the square root of the right side: 0.979795897 Break this problem into two subproblems by setting (y + 0.8) equal to 0.979795897 and -0.979795897.Subproblem 1
y + 0.8 = 0.979795897 Simplifying y + 0.8 = 0.979795897 Reorder the terms: 0.8 + y = 0.979795897 Solving 0.8 + y = 0.979795897 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + y = 0.979795897 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + y = 0.979795897 + -0.8 y = 0.979795897 + -0.8 Combine like terms: 0.979795897 + -0.8 = 0.179795897 y = 0.179795897 Simplifying y = 0.179795897Subproblem 2
y + 0.8 = -0.979795897 Simplifying y + 0.8 = -0.979795897 Reorder the terms: 0.8 + y = -0.979795897 Solving 0.8 + y = -0.979795897 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + y = -0.979795897 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + y = -0.979795897 + -0.8 y = -0.979795897 + -0.8 Combine like terms: -0.979795897 + -0.8 = -1.779795897 y = -1.779795897 Simplifying y = -1.779795897Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.179795897, -1.779795897}
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