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Simplifying 20 + -3y(2 + 2y) = 1 20 + (2 * -3y + 2y * -3y) = 1 20 + (-6y + -6y2) = 1 Solving 20 + -6y + -6y2 = 1 Solving for variable 'y'. Reorder the terms: 20 + -1 + -6y + -6y2 = 1 + -1 Combine like terms: 20 + -1 = 19 19 + -6y + -6y2 = 1 + -1 Combine like terms: 1 + -1 = 0 19 + -6y + -6y2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -3.166666667 + y + y2 = 0 Move the constant term to the right: Add '3.166666667' to each side of the equation. -3.166666667 + y + 3.166666667 + y2 = 0 + 3.166666667 Reorder the terms: -3.166666667 + 3.166666667 + y + y2 = 0 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + y + y2 = 0 + 3.166666667 y + y2 = 0 + 3.166666667 Combine like terms: 0 + 3.166666667 = 3.166666667 y + y2 = 3.166666667 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. + 0.25 + y2 = 3.166666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 3.166666667 + 0.25 Combine like terms: 3.166666667 + 0.25 = 3.416666667 1.25 + y2 = 3.416666667 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 3.416666667 Calculate the square root of the right side: 1.848422751 Break this problem into two subproblems by setting (y + 0.5) equal to 1.848422751 and -1.848422751.Subproblem 1
y + 0.5 = 1.848422751 Simplifying y + 0.5 = 1.848422751 Reorder the terms: 0.5 + y = 1.848422751 Solving 0.5 + y = 1.848422751 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 = 1.848422751 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.848422751 + -0.5 y = 1.848422751 + -0.5 Combine like terms: 1.848422751 + -0.5 = 1.348422751 y = 1.348422751 Simplifying y = 1.348422751Subproblem 2
y + 0.5 = -1.848422751 Simplifying y + 0.5 = -1.848422751 Reorder the terms: 0.5 + y = -1.848422751 Solving 0.5 + y = -1.848422751 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 = -1.848422751 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.848422751 + -0.5 y = -1.848422751 + -0.5 Combine like terms: -1.848422751 + -0.5 = -2.348422751 y = -2.348422751 Simplifying y = -2.348422751Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.348422751, -2.348422751}
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