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Simplifying 2(3y + -2) * 3y = 5 Reorder the terms: 2(-2 + 3y) * 3y = 5 Reorder the terms for easier multiplication: 2 * 3y(-2 + 3y) = 5 Multiply 2 * 3 6y(-2 + 3y) = 5 (-2 * 6y + 3y * 6y) = 5 (-12y + 18y2) = 5 Solving -12y + 18y2 = 5 Solving for variable 'y'. Reorder the terms: -5 + -12y + 18y2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -12y + 18y2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -0.2777777778 + -0.6666666667y + y2 = 0 Move the constant term to the right: Add '0.2777777778' to each side of the equation. -0.2777777778 + -0.6666666667y + 0.2777777778 + y2 = 0 + 0.2777777778 Reorder the terms: -0.2777777778 + 0.2777777778 + -0.6666666667y + y2 = 0 + 0.2777777778 Combine like terms: -0.2777777778 + 0.2777777778 = 0.0000000000 0.0000000000 + -0.6666666667y + y2 = 0 + 0.2777777778 -0.6666666667y + y2 = 0 + 0.2777777778 Combine like terms: 0 + 0.2777777778 = 0.2777777778 -0.6666666667y + y2 = 0.2777777778 The y term is -0.6666666667y. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667y + 0.1111111112 + y2 = 0.2777777778 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667y + y2 = 0.2777777778 + 0.1111111112 Combine like terms: 0.2777777778 + 0.1111111112 = 0.388888889 0.1111111112 + -0.6666666667y + y2 = 0.388888889 Factor a perfect square on the left side: (y + -0.3333333334)(y + -0.3333333334) = 0.388888889 Calculate the square root of the right side: 0.623609565 Break this problem into two subproblems by setting (y + -0.3333333334) equal to 0.623609565 and -0.623609565.Subproblem 1
y + -0.3333333334 = 0.623609565 Simplifying y + -0.3333333334 = 0.623609565 Reorder the terms: -0.3333333334 + y = 0.623609565 Solving -0.3333333334 + y = 0.623609565 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = 0.623609565 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 0.623609565 + 0.3333333334 y = 0.623609565 + 0.3333333334 Combine like terms: 0.623609565 + 0.3333333334 = 0.9569428984 y = 0.9569428984 Simplifying y = 0.9569428984Subproblem 2
y + -0.3333333334 = -0.623609565 Simplifying y + -0.3333333334 = -0.623609565 Reorder the terms: -0.3333333334 + y = -0.623609565 Solving -0.3333333334 + y = -0.623609565 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + y = -0.623609565 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -0.623609565 + 0.3333333334 y = -0.623609565 + 0.3333333334 Combine like terms: -0.623609565 + 0.3333333334 = -0.2902762316 y = -0.2902762316 Simplifying y = -0.2902762316Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.9569428984, -0.2902762316}
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