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Simplifying 30y3 + -45y2 + -3y = 3y Reorder the terms: -3y + -45y2 + 30y3 = 3y Solving -3y + -45y2 + 30y3 = 3y Solving for variable 'y'. Reorder the terms: -3y + -3y + -45y2 + 30y3 = 3y + -3y Combine like terms: -3y + -3y = -6y -6y + -45y2 + 30y3 = 3y + -3y Combine like terms: 3y + -3y = 0 -6y + -45y2 + 30y3 = 0 Factor out the Greatest Common Factor (GCF), '3y'. 3y(-2 + -15y + 10y2) = 0 Ignore the factor 3.Subproblem 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 '(-2 + -15y + 10y2)' equal to zero and attempt to solve: Simplifying -2 + -15y + 10y2 = 0 Solving -2 + -15y + 10y2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -0.2 + -1.5y + y2 = 0.0 Move the constant term to the right: Add '0.2' to each side of the equation. -0.2 + -1.5y + 0.2 + y2 = 0.0 + 0.2 Reorder the terms: -0.2 + 0.2 + -1.5y + y2 = 0.0 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + -1.5y + y2 = 0.0 + 0.2 -1.5y + y2 = 0.0 + 0.2 Combine like terms: 0.0 + 0.2 = 0.2 -1.5y + y2 = 0.2 The y term is -1.5y. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5y + 0.5625 + y2 = 0.2 + 0.5625 Reorder the terms: 0.5625 + -1.5y + y2 = 0.2 + 0.5625 Combine like terms: 0.2 + 0.5625 = 0.7625 0.5625 + -1.5y + y2 = 0.7625 Factor a perfect square on the left side: (y + -0.75)(y + -0.75) = 0.7625 Calculate the square root of the right side: 0.87321246 Break this problem into two subproblems by setting (y + -0.75) equal to 0.87321246 and -0.87321246.Subproblem 1
y + -0.75 = 0.87321246 Simplifying y + -0.75 = 0.87321246 Reorder the terms: -0.75 + y = 0.87321246 Solving -0.75 + y = 0.87321246 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 0.87321246 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 0.87321246 + 0.75 y = 0.87321246 + 0.75 Combine like terms: 0.87321246 + 0.75 = 1.62321246 y = 1.62321246 Simplifying y = 1.62321246Subproblem 2
y + -0.75 = -0.87321246 Simplifying y + -0.75 = -0.87321246 Reorder the terms: -0.75 + y = -0.87321246 Solving -0.75 + y = -0.87321246 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -0.87321246 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -0.87321246 + 0.75 y = -0.87321246 + 0.75 Combine like terms: -0.87321246 + 0.75 = -0.12321246 y = -0.12321246 Simplifying y = -0.12321246Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.62321246, -0.12321246}Solution
y = {0, 1.62321246, -0.12321246}
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