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Simplifying 21y + 14y2 = 1000 Solving 21y + 14y2 = 1000 Solving for variable 'y'. Reorder the terms: -1000 + 21y + 14y2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -1000 + 21y + 14y2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -71.42857143 + 1.5y + y2 = 0 Move the constant term to the right: Add '71.42857143' to each side of the equation. -71.42857143 + 1.5y + 71.42857143 + y2 = 0 + 71.42857143 Reorder the terms: -71.42857143 + 71.42857143 + 1.5y + y2 = 0 + 71.42857143 Combine like terms: -71.42857143 + 71.42857143 = 0.00000000 0.00000000 + 1.5y + y2 = 0 + 71.42857143 1.5y + y2 = 0 + 71.42857143 Combine like terms: 0 + 71.42857143 = 71.42857143 1.5y + y2 = 71.42857143 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = 71.42857143 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 71.42857143 + 0.5625 Combine like terms: 71.42857143 + 0.5625 = 71.99107143 0.5625 + 1.5y + y2 = 71.99107143 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 71.99107143 Calculate the square root of the right side: 8.484755237 Break this problem into two subproblems by setting (y + 0.75) equal to 8.484755237 and -8.484755237.Subproblem 1
y + 0.75 = 8.484755237 Simplifying y + 0.75 = 8.484755237 Reorder the terms: 0.75 + y = 8.484755237 Solving 0.75 + y = 8.484755237 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 8.484755237 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 8.484755237 + -0.75 y = 8.484755237 + -0.75 Combine like terms: 8.484755237 + -0.75 = 7.734755237 y = 7.734755237 Simplifying y = 7.734755237Subproblem 2
y + 0.75 = -8.484755237 Simplifying y + 0.75 = -8.484755237 Reorder the terms: 0.75 + y = -8.484755237 Solving 0.75 + y = -8.484755237 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -8.484755237 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -8.484755237 + -0.75 y = -8.484755237 + -0.75 Combine like terms: -8.484755237 + -0.75 = -9.234755237 y = -9.234755237 Simplifying y = -9.234755237Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.734755237, -9.234755237}
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