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Simplifying 3y2 + 10y + 6 = 0 Reorder the terms: 6 + 10y + 3y2 = 0 Solving 6 + 10y + 3y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2 + 3.333333333y + y2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 3.333333333y + -2 + y2 = 0 + -2 Reorder the terms: 2 + -2 + 3.333333333y + y2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 3.333333333y + y2 = 0 + -2 3.333333333y + y2 = 0 + -2 Combine like terms: 0 + -2 = -2 3.333333333y + y2 = -2 The y term is 3.333333333y. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333y + 2.777777779 + y2 = -2 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333y + y2 = -2 + 2.777777779 Combine like terms: -2 + 2.777777779 = 0.777777779 2.777777779 + 3.333333333y + y2 = 0.777777779 Factor a perfect square on the left side: (y + 1.666666667)(y + 1.666666667) = 0.777777779 Calculate the square root of the right side: 0.881917104 Break this problem into two subproblems by setting (y + 1.666666667) equal to 0.881917104 and -0.881917104.Subproblem 1
y + 1.666666667 = 0.881917104 Simplifying y + 1.666666667 = 0.881917104 Reorder the terms: 1.666666667 + y = 0.881917104 Solving 1.666666667 + y = 0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = 0.881917104 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = 0.881917104 + -1.666666667 y = 0.881917104 + -1.666666667 Combine like terms: 0.881917104 + -1.666666667 = -0.784749563 y = -0.784749563 Simplifying y = -0.784749563Subproblem 2
y + 1.666666667 = -0.881917104 Simplifying y + 1.666666667 = -0.881917104 Reorder the terms: 1.666666667 + y = -0.881917104 Solving 1.666666667 + y = -0.881917104 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + y = -0.881917104 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + y = -0.881917104 + -1.666666667 y = -0.881917104 + -1.666666667 Combine like terms: -0.881917104 + -1.666666667 = -2.548583771 y = -2.548583771 Simplifying y = -2.548583771Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.784749563, -2.548583771}
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