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Simplifying 3(7y + 2) = y(7 + 2y) + -1 Reorder the terms: 3(2 + 7y) = y(7 + 2y) + -1 (2 * 3 + 7y * 3) = y(7 + 2y) + -1 (6 + 21y) = y(7 + 2y) + -1 6 + 21y = (7 * y + 2y * y) + -1 6 + 21y = (7y + 2y2) + -1 Reorder the terms: 6 + 21y = -1 + 7y + 2y2 Solving 6 + 21y = -1 + 7y + 2y2 Solving for variable 'y'. Reorder the terms: 6 + 1 + 21y + -7y + -2y2 = -1 + 7y + 2y2 + 1 + -7y + -2y2 Combine like terms: 6 + 1 = 7 7 + 21y + -7y + -2y2 = -1 + 7y + 2y2 + 1 + -7y + -2y2 Combine like terms: 21y + -7y = 14y 7 + 14y + -2y2 = -1 + 7y + 2y2 + 1 + -7y + -2y2 Reorder the terms: 7 + 14y + -2y2 = -1 + 1 + 7y + -7y + 2y2 + -2y2 Combine like terms: -1 + 1 = 0 7 + 14y + -2y2 = 0 + 7y + -7y + 2y2 + -2y2 7 + 14y + -2y2 = 7y + -7y + 2y2 + -2y2 Combine like terms: 7y + -7y = 0 7 + 14y + -2y2 = 0 + 2y2 + -2y2 7 + 14y + -2y2 = 2y2 + -2y2 Combine like terms: 2y2 + -2y2 = 0 7 + 14y + -2y2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -3.5 + -7y + y2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -7y + 3.5 + y2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -7y + y2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -7y + y2 = 0 + 3.5 -7y + y2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -7y + y2 = 3.5 The y term is -7y. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7y + 12.25 + y2 = 3.5 + 12.25 Reorder the terms: 12.25 + -7y + y2 = 3.5 + 12.25 Combine like terms: 3.5 + 12.25 = 15.75 12.25 + -7y + y2 = 15.75 Factor a perfect square on the left side: (y + -3.5)(y + -3.5) = 15.75 Calculate the square root of the right side: 3.968626967 Break this problem into two subproblems by setting (y + -3.5) equal to 3.968626967 and -3.968626967.Subproblem 1
y + -3.5 = 3.968626967 Simplifying y + -3.5 = 3.968626967 Reorder the terms: -3.5 + y = 3.968626967 Solving -3.5 + y = 3.968626967 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + y = 3.968626967 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + y = 3.968626967 + 3.5 y = 3.968626967 + 3.5 Combine like terms: 3.968626967 + 3.5 = 7.468626967 y = 7.468626967 Simplifying y = 7.468626967Subproblem 2
y + -3.5 = -3.968626967 Simplifying y + -3.5 = -3.968626967 Reorder the terms: -3.5 + y = -3.968626967 Solving -3.5 + y = -3.968626967 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + y = -3.968626967 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + y = -3.968626967 + 3.5 y = -3.968626967 + 3.5 Combine like terms: -3.968626967 + 3.5 = -0.468626967 y = -0.468626967 Simplifying y = -0.468626967Solution
The solution to the problem is based on the solutions from the subproblems. y = {7.468626967, -0.468626967}
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