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Simplifying 12y2 + 16y = -3 Reorder the terms: 16y + 12y2 = -3 Solving 16y + 12y2 = -3 Solving for variable 'y'. Reorder the terms: 3 + 16y + 12y2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + 16y + 12y2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.25 + 1.333333333y + y2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + 1.333333333y + -0.25 + y2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + 1.333333333y + y2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + 1.333333333y + y2 = 0 + -0.25 1.333333333y + y2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 1.333333333y + y2 = -0.25 The y term is 1.333333333y. Take half its coefficient (0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. 1.333333333y + 0.4444444442 + y2 = -0.25 + 0.4444444442 Reorder the terms: 0.4444444442 + 1.333333333y + y2 = -0.25 + 0.4444444442 Combine like terms: -0.25 + 0.4444444442 = 0.1944444442 0.4444444442 + 1.333333333y + y2 = 0.1944444442 Factor a perfect square on the left side: (y + 0.6666666665)(y + 0.6666666665) = 0.1944444442 Calculate the square root of the right side: 0.440958552 Break this problem into two subproblems by setting (y + 0.6666666665) equal to 0.440958552 and -0.440958552.Subproblem 1
y + 0.6666666665 = 0.440958552 Simplifying y + 0.6666666665 = 0.440958552 Reorder the terms: 0.6666666665 + y = 0.440958552 Solving 0.6666666665 + y = 0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = 0.440958552 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = 0.440958552 + -0.6666666665 y = 0.440958552 + -0.6666666665 Combine like terms: 0.440958552 + -0.6666666665 = -0.2257081145 y = -0.2257081145 Simplifying y = -0.2257081145Subproblem 2
y + 0.6666666665 = -0.440958552 Simplifying y + 0.6666666665 = -0.440958552 Reorder the terms: 0.6666666665 + y = -0.440958552 Solving 0.6666666665 + y = -0.440958552 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.6666666665' to each side of the equation. 0.6666666665 + -0.6666666665 + y = -0.440958552 + -0.6666666665 Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000 0.0000000000 + y = -0.440958552 + -0.6666666665 y = -0.440958552 + -0.6666666665 Combine like terms: -0.440958552 + -0.6666666665 = -1.1076252185 y = -1.1076252185 Simplifying y = -1.1076252185Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.2257081145, -1.1076252185}
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