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Simplifying 3y(2y + 3) + 4y2 + 11y + 5 = 0 Reorder the terms: 3y(3 + 2y) + 4y2 + 11y + 5 = 0 (3 * 3y + 2y * 3y) + 4y2 + 11y + 5 = 0 (9y + 6y2) + 4y2 + 11y + 5 = 0 Reorder the terms: 5 + 9y + 11y + 6y2 + 4y2 = 0 Combine like terms: 9y + 11y = 20y 5 + 20y + 6y2 + 4y2 = 0 Combine like terms: 6y2 + 4y2 = 10y2 5 + 20y + 10y2 = 0 Solving 5 + 20y + 10y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '5'. 5(1 + 4y + 2y2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(1 + 4y + 2y2)' equal to zero and attempt to solve: Simplifying 1 + 4y + 2y2 = 0 Solving 1 + 4y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + 2y + y2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 2y + -0.5 + y2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 2y + y2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 2y + y2 = 0 + -0.5 2y + y2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 2y + y2 = -0.5 The y term is 2y. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2y + 1 + y2 = -0.5 + 1 Reorder the terms: 1 + 2y + y2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + 2y + y2 = 0.5 Factor a perfect square on the left side: (y + 1)(y + 1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (y + 1) equal to 0.707106781 and -0.707106781.Subproblem 1
y + 1 = 0.707106781 Simplifying y + 1 = 0.707106781 Reorder the terms: 1 + y = 0.707106781 Solving 1 + y = 0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = 0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + y = 0.707106781 + -1 y = 0.707106781 + -1 Combine like terms: 0.707106781 + -1 = -0.292893219 y = -0.292893219 Simplifying y = -0.292893219Subproblem 2
y + 1 = -0.707106781 Simplifying y + 1 = -0.707106781 Reorder the terms: 1 + y = -0.707106781 Solving 1 + y = -0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + y = -0.707106781 + -1 Combine like terms: 1 + -1 = 0 0 + y = -0.707106781 + -1 y = -0.707106781 + -1 Combine like terms: -0.707106781 + -1 = -1.707106781 y = -1.707106781 Simplifying y = -1.707106781Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.292893219, -1.707106781}Solution
y = {-0.292893219, -1.707106781}
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