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Simplifying 3y(2y + 3) = 10 Reorder the terms: 3y(3 + 2y) = 10 (3 * 3y + 2y * 3y) = 10 (9y + 6y2) = 10 Solving 9y + 6y2 = 10 Solving for variable 'y'. Reorder the terms: -10 + 9y + 6y2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 9y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.666666667 + 1.5y + y2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 1.5y + 1.666666667 + y2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 1.5y + y2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 1.5y + y2 = 0 + 1.666666667 1.5y + y2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 1.5y + y2 = 1.666666667 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = 1.666666667 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 1.666666667 + 0.5625 Combine like terms: 1.666666667 + 0.5625 = 2.229166667 0.5625 + 1.5y + y2 = 2.229166667 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 2.229166667 Calculate the square root of the right side: 1.493039406 Break this problem into two subproblems by setting (y + 0.75) equal to 1.493039406 and -1.493039406.Subproblem 1
y + 0.75 = 1.493039406 Simplifying y + 0.75 = 1.493039406 Reorder the terms: 0.75 + y = 1.493039406 Solving 0.75 + y = 1.493039406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 1.493039406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 1.493039406 + -0.75 y = 1.493039406 + -0.75 Combine like terms: 1.493039406 + -0.75 = 0.743039406 y = 0.743039406 Simplifying y = 0.743039406Subproblem 2
y + 0.75 = -1.493039406 Simplifying y + 0.75 = -1.493039406 Reorder the terms: 0.75 + y = -1.493039406 Solving 0.75 + y = -1.493039406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -1.493039406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -1.493039406 + -0.75 y = -1.493039406 + -0.75 Combine like terms: -1.493039406 + -0.75 = -2.243039406 y = -2.243039406 Simplifying y = -2.243039406Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.743039406, -2.243039406}
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