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Simplifying 12y2 + 24y = 5 Reorder the terms: 24y + 12y2 = 5 Solving 24y + 12y2 = 5 Solving for variable 'y'. Reorder the terms: -5 + 24y + 12y2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + 24y + 12y2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + 2y + y2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + 2y + 0.4166666667 + y2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + 2y + y2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + 2y + y2 = 0 + 0.4166666667 2y + y2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 2y + y2 = 0.4166666667 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = 0.4166666667 + 1 Reorder the terms: 1 + 2y + y2 = 0.4166666667 + 1 Combine like terms: 0.4166666667 + 1 = 1.4166666667 1 + 2y + y2 = 1.4166666667 Factor a perfect square on the left side: (y + 1)(y + 1) = 1.4166666667 Calculate the square root of the right side: 1.190238071 Break this problem into two subproblems by setting (y + 1) equal to 1.190238071 and -1.190238071.Subproblem 1
y + 1 = 1.190238071 Simplifying y + 1 = 1.190238071 Reorder the terms: 1 + y = 1.190238071 Solving 1 + y = 1.190238071 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 1.190238071 + -1 Combine like terms: 1 + -1 = 0 0 + y = 1.190238071 + -1 y = 1.190238071 + -1 Combine like terms: 1.190238071 + -1 = 0.190238071 y = 0.190238071 Simplifying y = 0.190238071Subproblem 2
y + 1 = -1.190238071 Simplifying y + 1 = -1.190238071 Reorder the terms: 1 + y = -1.190238071 Solving 1 + y = -1.190238071 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -1.190238071 + -1 Combine like terms: 1 + -1 = 0 0 + y = -1.190238071 + -1 y = -1.190238071 + -1 Combine like terms: -1.190238071 + -1 = -2.190238071 y = -2.190238071 Simplifying y = -2.190238071Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.190238071, -2.190238071}
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