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Simplifying 0.19y + 0.05y(y + 3000) = 1590 Reorder the terms: 0.19y + 0.05y(3000 + y) = 1590 0.19y + (3000 * 0.05y + y * 0.05y) = 1590 0.19y + (150y + 0.05y2) = 1590 Combine like terms: 0.19y + 150y = 150.19y 150.19y + 0.05y2 = 1590 Solving 150.19y + 0.05y2 = 1590 Solving for variable 'y'. Reorder the terms: -1590 + 150.19y + 0.05y2 = 1590 + -1590 Combine like terms: 1590 + -1590 = 0 -1590 + 150.19y + 0.05y2 = 0 Begin completing the square. Divide all terms by 0.05 the coefficient of the squared term: Divide each side by '0.05'. -31800 + 3003.8y + y2 = 0 Move the constant term to the right: Add '31800' to each side of the equation. -31800 + 3003.8y + 31800 + y2 = 0 + 31800 Reorder the terms: -31800 + 31800 + 3003.8y + y2 = 0 + 31800 Combine like terms: -31800 + 31800 = 0 0 + 3003.8y + y2 = 0 + 31800 3003.8y + y2 = 0 + 31800 Combine like terms: 0 + 31800 = 31800 3003.8y + y2 = 31800 The y term is 3003.8y. Take half its coefficient (1501.9). Square it (2255703.61) and add it to both sides. Add '2255703.61' to each side of the equation. 3003.8y + 2255703.61 + y2 = 31800 + 2255703.61 Reorder the terms: 2255703.61 + 3003.8y + y2 = 31800 + 2255703.61 Combine like terms: 31800 + 2255703.61 = 2287503.61 2255703.61 + 3003.8y + y2 = 2287503.61 Factor a perfect square on the left side: (y + 1501.9)(y + 1501.9) = 2287503.61 Calculate the square root of the right side: 1512.449539654 Break this problem into two subproblems by setting (y + 1501.9) equal to 1512.449539654 and -1512.449539654.Subproblem 1
y + 1501.9 = 1512.449539654 Simplifying y + 1501.9 = 1512.449539654 Reorder the terms: 1501.9 + y = 1512.449539654 Solving 1501.9 + y = 1512.449539654 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1501.9' to each side of the equation. 1501.9 + -1501.9 + y = 1512.449539654 + -1501.9 Combine like terms: 1501.9 + -1501.9 = 0.0 0.0 + y = 1512.449539654 + -1501.9 y = 1512.449539654 + -1501.9 Combine like terms: 1512.449539654 + -1501.9 = 10.549539654 y = 10.549539654 Simplifying y = 10.549539654Subproblem 2
y + 1501.9 = -1512.449539654 Simplifying y + 1501.9 = -1512.449539654 Reorder the terms: 1501.9 + y = -1512.449539654 Solving 1501.9 + y = -1512.449539654 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1501.9' to each side of the equation. 1501.9 + -1501.9 + y = -1512.449539654 + -1501.9 Combine like terms: 1501.9 + -1501.9 = 0.0 0.0 + y = -1512.449539654 + -1501.9 y = -1512.449539654 + -1501.9 Combine like terms: -1512.449539654 + -1501.9 = -3014.349539654 y = -3014.349539654 Simplifying y = -3014.349539654Solution
The solution to the problem is based on the solutions from the subproblems. y = {10.549539654, -3014.349539654}
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