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Simplifying 54y2 + 291y + 30 = 0 Reorder the terms: 30 + 291y + 54y2 = 0 Solving 30 + 291y + 54y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3'. 3(10 + 97y + 18y2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(10 + 97y + 18y2)' equal to zero and attempt to solve: Simplifying 10 + 97y + 18y2 = 0 Solving 10 + 97y + 18y2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.5555555556 + 5.388888889y + y2 = 0 Move the constant term to the right: Add '-0.5555555556' to each side of the equation. 0.5555555556 + 5.388888889y + -0.5555555556 + y2 = 0 + -0.5555555556 Reorder the terms: 0.5555555556 + -0.5555555556 + 5.388888889y + y2 = 0 + -0.5555555556 Combine like terms: 0.5555555556 + -0.5555555556 = 0.0000000000 0.0000000000 + 5.388888889y + y2 = 0 + -0.5555555556 5.388888889y + y2 = 0 + -0.5555555556 Combine like terms: 0 + -0.5555555556 = -0.5555555556 5.388888889y + y2 = -0.5555555556 The y term is 5.388888889y. Take half its coefficient (2.694444445). Square it (7.260030867) and add it to both sides. Add '7.260030867' to each side of the equation. 5.388888889y + 7.260030867 + y2 = -0.5555555556 + 7.260030867 Reorder the terms: 7.260030867 + 5.388888889y + y2 = -0.5555555556 + 7.260030867 Combine like terms: -0.5555555556 + 7.260030867 = 6.7044753114 7.260030867 + 5.388888889y + y2 = 6.7044753114 Factor a perfect square on the left side: (y + 2.694444445)(y + 2.694444445) = 6.7044753114 Calculate the square root of the right side: 2.589300159 Break this problem into two subproblems by setting (y + 2.694444445) equal to 2.589300159 and -2.589300159.Subproblem 1
y + 2.694444445 = 2.589300159 Simplifying y + 2.694444445 = 2.589300159 Reorder the terms: 2.694444445 + y = 2.589300159 Solving 2.694444445 + y = 2.589300159 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.694444445' to each side of the equation. 2.694444445 + -2.694444445 + y = 2.589300159 + -2.694444445 Combine like terms: 2.694444445 + -2.694444445 = 0.000000000 0.000000000 + y = 2.589300159 + -2.694444445 y = 2.589300159 + -2.694444445 Combine like terms: 2.589300159 + -2.694444445 = -0.105144286 y = -0.105144286 Simplifying y = -0.105144286Subproblem 2
y + 2.694444445 = -2.589300159 Simplifying y + 2.694444445 = -2.589300159 Reorder the terms: 2.694444445 + y = -2.589300159 Solving 2.694444445 + y = -2.589300159 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-2.694444445' to each side of the equation. 2.694444445 + -2.694444445 + y = -2.589300159 + -2.694444445 Combine like terms: 2.694444445 + -2.694444445 = 0.000000000 0.000000000 + y = -2.589300159 + -2.694444445 y = -2.589300159 + -2.694444445 Combine like terms: -2.589300159 + -2.694444445 = -5.283744604 y = -5.283744604 Simplifying y = -5.283744604Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.105144286, -5.283744604}Solution
y = {-0.105144286, -5.283744604}
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