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Simplifying 4y2 + 112y + 180 = 0 Reorder the terms: 180 + 112y + 4y2 = 0 Solving 180 + 112y + 4y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '4'. 4(45 + 28y + y2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(45 + 28y + y2)' equal to zero and attempt to solve: Simplifying 45 + 28y + y2 = 0 Solving 45 + 28y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-45' to each side of the equation. 45 + 28y + -45 + y2 = 0 + -45 Reorder the terms: 45 + -45 + 28y + y2 = 0 + -45 Combine like terms: 45 + -45 = 0 0 + 28y + y2 = 0 + -45 28y + y2 = 0 + -45 Combine like terms: 0 + -45 = -45 28y + y2 = -45 The y term is 28y. Take half its coefficient (14). Square it (196) and add it to both sides. Add '196' to each side of the equation. 28y + 196 + y2 = -45 + 196 Reorder the terms: 196 + 28y + y2 = -45 + 196 Combine like terms: -45 + 196 = 151 196 + 28y + y2 = 151 Factor a perfect square on the left side: (y + 14)(y + 14) = 151 Calculate the square root of the right side: 12.288205727 Break this problem into two subproblems by setting (y + 14) equal to 12.288205727 and -12.288205727.Subproblem 1
y + 14 = 12.288205727 Simplifying y + 14 = 12.288205727 Reorder the terms: 14 + y = 12.288205727 Solving 14 + y = 12.288205727 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + y = 12.288205727 + -14 Combine like terms: 14 + -14 = 0 0 + y = 12.288205727 + -14 y = 12.288205727 + -14 Combine like terms: 12.288205727 + -14 = -1.711794273 y = -1.711794273 Simplifying y = -1.711794273Subproblem 2
y + 14 = -12.288205727 Simplifying y + 14 = -12.288205727 Reorder the terms: 14 + y = -12.288205727 Solving 14 + y = -12.288205727 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + y = -12.288205727 + -14 Combine like terms: 14 + -14 = 0 0 + y = -12.288205727 + -14 y = -12.288205727 + -14 Combine like terms: -12.288205727 + -14 = -26.288205727 y = -26.288205727 Simplifying y = -26.288205727Solution
The solution to the problem is based on the solutions from the subproblems. y = {-1.711794273, -26.288205727}Solution
y = {-1.711794273, -26.288205727}
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