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Simplifying 6y2 + -4y = 3 Reorder the terms: -4y + 6y2 = 3 Solving -4y + 6y2 = 3 Solving for variable 'y'. Reorder the terms: -3 + -4y + 6y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -4y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + -0.6666666667y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -0.6666666667y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -0.6666666667y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -0.6666666667y + y2 = 0 + 0.5 -0.6666666667y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -0.6666666667y + y2 = 0.5 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 = 0.5 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667y + y2 = 0.5 + 0.1111111112 Combine like terms: 0.5 + 0.1111111112 = 0.6111111112 0.1111111112 + -0.6666666667y + y2 = 0.6111111112 Factor a perfect square on the left side: (y + -0.3333333334)(y + -0.3333333334) = 0.6111111112 Calculate the square root of the right side: 0.78173596 Break this problem into two subproblems by setting (y + -0.3333333334) equal to 0.78173596 and -0.78173596.Subproblem 1
y + -0.3333333334 = 0.78173596 Simplifying y + -0.3333333334 = 0.78173596 Reorder the terms: -0.3333333334 + y = 0.78173596 Solving -0.3333333334 + y = 0.78173596 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 = 0.78173596 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = 0.78173596 + 0.3333333334 y = 0.78173596 + 0.3333333334 Combine like terms: 0.78173596 + 0.3333333334 = 1.1150692934 y = 1.1150692934 Simplifying y = 1.1150692934Subproblem 2
y + -0.3333333334 = -0.78173596 Simplifying y + -0.3333333334 = -0.78173596 Reorder the terms: -0.3333333334 + y = -0.78173596 Solving -0.3333333334 + y = -0.78173596 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 = -0.78173596 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + y = -0.78173596 + 0.3333333334 y = -0.78173596 + 0.3333333334 Combine like terms: -0.78173596 + 0.3333333334 = -0.4484026266 y = -0.4484026266 Simplifying y = -0.4484026266Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.1150692934, -0.4484026266}
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