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Simplifying 4y2 + -32y + 64 + y2 = y2 + 8y + 16 Reorder the terms: 64 + -32y + 4y2 + y2 = y2 + 8y + 16 Combine like terms: 4y2 + y2 = 5y2 64 + -32y + 5y2 = y2 + 8y + 16 Reorder the terms: 64 + -32y + 5y2 = 16 + 8y + y2 Solving 64 + -32y + 5y2 = 16 + 8y + y2 Solving for variable 'y'. Reorder the terms: 64 + -16 + -32y + -8y + 5y2 + -1y2 = 16 + 8y + y2 + -16 + -8y + -1y2 Combine like terms: 64 + -16 = 48 48 + -32y + -8y + 5y2 + -1y2 = 16 + 8y + y2 + -16 + -8y + -1y2 Combine like terms: -32y + -8y = -40y 48 + -40y + 5y2 + -1y2 = 16 + 8y + y2 + -16 + -8y + -1y2 Combine like terms: 5y2 + -1y2 = 4y2 48 + -40y + 4y2 = 16 + 8y + y2 + -16 + -8y + -1y2 Reorder the terms: 48 + -40y + 4y2 = 16 + -16 + 8y + -8y + y2 + -1y2 Combine like terms: 16 + -16 = 0 48 + -40y + 4y2 = 0 + 8y + -8y + y2 + -1y2 48 + -40y + 4y2 = 8y + -8y + y2 + -1y2 Combine like terms: 8y + -8y = 0 48 + -40y + 4y2 = 0 + y2 + -1y2 48 + -40y + 4y2 = y2 + -1y2 Combine like terms: y2 + -1y2 = 0 48 + -40y + 4y2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(12 + -10y + y2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(12 + -10y + y2)' equal to zero and attempt to solve: Simplifying 12 + -10y + y2 = 0 Solving 12 + -10y + y2 = 0 Begin completing the square. Move the constant term to the right: Add '-12' to each side of the equation. 12 + -10y + -12 + y2 = 0 + -12 Reorder the terms: 12 + -12 + -10y + y2 = 0 + -12 Combine like terms: 12 + -12 = 0 0 + -10y + y2 = 0 + -12 -10y + y2 = 0 + -12 Combine like terms: 0 + -12 = -12 -10y + y2 = -12 The y term is -10y. Take half its coefficient (-5). Square it (25) and add it to both sides. Add '25' to each side of the equation. -10y + 25 + y2 = -12 + 25 Reorder the terms: 25 + -10y + y2 = -12 + 25 Combine like terms: -12 + 25 = 13 25 + -10y + y2 = 13 Factor a perfect square on the left side: (y + -5)(y + -5) = 13 Calculate the square root of the right side: 3.605551275 Break this problem into two subproblems by setting (y + -5) equal to 3.605551275 and -3.605551275.Subproblem 1
y + -5 = 3.605551275 Simplifying y + -5 = 3.605551275 Reorder the terms: -5 + y = 3.605551275 Solving -5 + y = 3.605551275 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = 3.605551275 + 5 Combine like terms: -5 + 5 = 0 0 + y = 3.605551275 + 5 y = 3.605551275 + 5 Combine like terms: 3.605551275 + 5 = 8.605551275 y = 8.605551275 Simplifying y = 8.605551275Subproblem 2
y + -5 = -3.605551275 Simplifying y + -5 = -3.605551275 Reorder the terms: -5 + y = -3.605551275 Solving -5 + y = -3.605551275 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + y = -3.605551275 + 5 Combine like terms: -5 + 5 = 0 0 + y = -3.605551275 + 5 y = -3.605551275 + 5 Combine like terms: -3.605551275 + 5 = 1.394448725 y = 1.394448725 Simplifying y = 1.394448725Solution
The solution to the problem is based on the solutions from the subproblems. y = {8.605551275, 1.394448725}Solution
y = {8.605551275, 1.394448725}
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