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Simplifying 3y2 + 5y = 3 Reorder the terms: 5y + 3y2 = 3 Solving 5y + 3y2 = 3 Solving for variable 'y'. Reorder the terms: -3 + 5y + 3y2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 5y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + 1.666666667y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.666666667y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + 1.666666667y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.666666667y + y2 = 0 + 1 1.666666667y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.666666667y + y2 = 1 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 = 1 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667y + y2 = 1 + 0.6944444447 Combine like terms: 1 + 0.6944444447 = 1.6944444447 0.6944444447 + 1.666666667y + y2 = 1.6944444447 Factor a perfect square on the left side: (y + 0.8333333335)(y + 0.8333333335) = 1.6944444447 Calculate the square root of the right side: 1.301708279 Break this problem into two subproblems by setting (y + 0.8333333335) equal to 1.301708279 and -1.301708279.Subproblem 1
y + 0.8333333335 = 1.301708279 Simplifying y + 0.8333333335 = 1.301708279 Reorder the terms: 0.8333333335 + y = 1.301708279 Solving 0.8333333335 + y = 1.301708279 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 = 1.301708279 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = 1.301708279 + -0.8333333335 y = 1.301708279 + -0.8333333335 Combine like terms: 1.301708279 + -0.8333333335 = 0.4683749455 y = 0.4683749455 Simplifying y = 0.4683749455Subproblem 2
y + 0.8333333335 = -1.301708279 Simplifying y + 0.8333333335 = -1.301708279 Reorder the terms: 0.8333333335 + y = -1.301708279 Solving 0.8333333335 + y = -1.301708279 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 = -1.301708279 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + y = -1.301708279 + -0.8333333335 y = -1.301708279 + -0.8333333335 Combine like terms: -1.301708279 + -0.8333333335 = -2.1350416125 y = -2.1350416125 Simplifying y = -2.1350416125Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.4683749455, -2.1350416125}
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