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Simplifying 2(-3y + 5) * 3y = 7 Reorder the terms: 2(5 + -3y) * 3y = 7 Reorder the terms for easier multiplication: 2 * 3y(5 + -3y) = 7 Multiply 2 * 3 6y(5 + -3y) = 7 (5 * 6y + -3y * 6y) = 7 (30y + -18y2) = 7 Solving 30y + -18y2 = 7 Solving for variable 'y'. Reorder the terms: -7 + 30y + -18y2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 30y + -18y2 = 0 Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. 0.3888888889 + -1.666666667y + y2 = 0 Move the constant term to the right: Add '-0.3888888889' to each side of the equation. 0.3888888889 + -1.666666667y + -0.3888888889 + y2 = 0 + -0.3888888889 Reorder the terms: 0.3888888889 + -0.3888888889 + -1.666666667y + y2 = 0 + -0.3888888889 Combine like terms: 0.3888888889 + -0.3888888889 = 0.0000000000 0.0000000000 + -1.666666667y + y2 = 0 + -0.3888888889 -1.666666667y + y2 = 0 + -0.3888888889 Combine like terms: 0 + -0.3888888889 = -0.3888888889 -1.666666667y + y2 = -0.3888888889 The y term is -1.666666667y. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667y + 0.6944444447 + y2 = -0.3888888889 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667y + y2 = -0.3888888889 + 0.6944444447 Combine like terms: -0.3888888889 + 0.6944444447 = 0.3055555558 0.6944444447 + -1.666666667y + y2 = 0.3055555558 Factor a perfect square on the left side: (y + -0.8333333335)(y + -0.8333333335) = 0.3055555558 Calculate the square root of the right side: 0.552770799 Break this problem into two subproblems by setting (y + -0.8333333335) equal to 0.552770799 and -0.552770799.Subproblem 1
y + -0.8333333335 = 0.552770799 Simplifying y + -0.8333333335 = 0.552770799 Reorder the terms: -0.8333333335 + y = 0.552770799 Solving -0.8333333335 + y = 0.552770799 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = 0.552770799 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = 0.552770799 + 0.8333333335 y = 0.552770799 + 0.8333333335 Combine like terms: 0.552770799 + 0.8333333335 = 1.3861041325 y = 1.3861041325 Simplifying y = 1.3861041325Subproblem 2
y + -0.8333333335 = -0.552770799 Simplifying y + -0.8333333335 = -0.552770799 Reorder the terms: -0.8333333335 + y = -0.552770799 Solving -0.8333333335 + y = -0.552770799 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + y = -0.552770799 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + y = -0.552770799 + 0.8333333335 y = -0.552770799 + 0.8333333335 Combine like terms: -0.552770799 + 0.8333333335 = 0.2805625345 y = 0.2805625345 Simplifying y = 0.2805625345Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.3861041325, 0.2805625345}
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