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Simplifying (6y3 + 4y2) + -1(2y2 + 3y) = 0 Reorder the terms: (4y2 + 6y3) + -1(2y2 + 3y) = 0 Remove parenthesis around (4y2 + 6y3) 4y2 + 6y3 + -1(2y2 + 3y) = 0 Reorder the terms: 4y2 + 6y3 + -1(3y + 2y2) = 0 4y2 + 6y3 + (3y * -1 + 2y2 * -1) = 0 4y2 + 6y3 + (-3y + -2y2) = 0 Reorder the terms: -3y + 4y2 + -2y2 + 6y3 = 0 Combine like terms: 4y2 + -2y2 = 2y2 -3y + 2y2 + 6y3 = 0 Solving -3y + 2y2 + 6y3 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), 'y'. y(-3 + 2y + 6y2) = 0Subproblem 1
Set the factor 'y' equal to zero and attempt to solve: Simplifying y = 0 Solving y = 0 Move all terms containing y to the left, all other terms to the right. Simplifying y = 0Subproblem 2
Set the factor '(-3 + 2y + 6y2)' equal to zero and attempt to solve: Simplifying -3 + 2y + 6y2 = 0 Solving -3 + 2y + 6y2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.3333333333y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.3333333333y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.3333333333y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.3333333333y + y2 = 0 + 0.5 0.3333333333y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.3333333333y + y2 = 0.5 The y term is 0.3333333333y. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333y + 0.02777777779 + y2 = 0.5 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333y + y2 = 0.5 + 0.02777777779 Combine like terms: 0.5 + 0.02777777779 = 0.52777777779 0.02777777779 + 0.3333333333y + y2 = 0.52777777779 Factor a perfect square on the left side: (y + 0.1666666667)(y + 0.1666666667) = 0.52777777779 Calculate the square root of the right side: 0.726483157 Break this problem into two subproblems by setting (y + 0.1666666667) equal to 0.726483157 and -0.726483157.Subproblem 1
y + 0.1666666667 = 0.726483157 Simplifying y + 0.1666666667 = 0.726483157 Reorder the terms: 0.1666666667 + y = 0.726483157 Solving 0.1666666667 + y = 0.726483157 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = 0.726483157 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = 0.726483157 + -0.1666666667 y = 0.726483157 + -0.1666666667 Combine like terms: 0.726483157 + -0.1666666667 = 0.5598164903 y = 0.5598164903 Simplifying y = 0.5598164903Subproblem 2
y + 0.1666666667 = -0.726483157 Simplifying y + 0.1666666667 = -0.726483157 Reorder the terms: 0.1666666667 + y = -0.726483157 Solving 0.1666666667 + y = -0.726483157 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + y = -0.726483157 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + y = -0.726483157 + -0.1666666667 y = -0.726483157 + -0.1666666667 Combine like terms: -0.726483157 + -0.1666666667 = -0.8931498237 y = -0.8931498237 Simplifying y = -0.8931498237Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.5598164903, -0.8931498237}Solution
y = {0, 0.5598164903, -0.8931498237}
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