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Simplifying 2(y + 2) * y = 19 + -2(y + 3) Reorder the terms: 2(2 + y) * y = 19 + -2(y + 3) Reorder the terms for easier multiplication: 2y(2 + y) = 19 + -2(y + 3) (2 * 2y + y * 2y) = 19 + -2(y + 3) (4y + 2y2) = 19 + -2(y + 3) Reorder the terms: 4y + 2y2 = 19 + -2(3 + y) 4y + 2y2 = 19 + (3 * -2 + y * -2) 4y + 2y2 = 19 + (-6 + -2y) Combine like terms: 19 + -6 = 13 4y + 2y2 = 13 + -2y Solving 4y + 2y2 = 13 + -2y Solving for variable 'y'. Reorder the terms: -13 + 4y + 2y + 2y2 = 13 + -2y + -13 + 2y Combine like terms: 4y + 2y = 6y -13 + 6y + 2y2 = 13 + -2y + -13 + 2y Reorder the terms: -13 + 6y + 2y2 = 13 + -13 + -2y + 2y Combine like terms: 13 + -13 = 0 -13 + 6y + 2y2 = 0 + -2y + 2y -13 + 6y + 2y2 = -2y + 2y Combine like terms: -2y + 2y = 0 -13 + 6y + 2y2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -6.5 + 3y + y2 = 0 Move the constant term to the right: Add '6.5' to each side of the equation. -6.5 + 3y + 6.5 + y2 = 0 + 6.5 Reorder the terms: -6.5 + 6.5 + 3y + y2 = 0 + 6.5 Combine like terms: -6.5 + 6.5 = 0.0 0.0 + 3y + y2 = 0 + 6.5 3y + y2 = 0 + 6.5 Combine like terms: 0 + 6.5 = 6.5 3y + y2 = 6.5 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = 6.5 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 6.5 + 2.25 Combine like terms: 6.5 + 2.25 = 8.75 2.25 + 3y + y2 = 8.75 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 8.75 Calculate the square root of the right side: 2.958039892 Break this problem into two subproblems by setting (y + 1.5) equal to 2.958039892 and -2.958039892.Subproblem 1
y + 1.5 = 2.958039892 Simplifying y + 1.5 = 2.958039892 Reorder the terms: 1.5 + y = 2.958039892 Solving 1.5 + y = 2.958039892 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 2.958039892 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 2.958039892 + -1.5 y = 2.958039892 + -1.5 Combine like terms: 2.958039892 + -1.5 = 1.458039892 y = 1.458039892 Simplifying y = 1.458039892Subproblem 2
y + 1.5 = -2.958039892 Simplifying y + 1.5 = -2.958039892 Reorder the terms: 1.5 + y = -2.958039892 Solving 1.5 + y = -2.958039892 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -2.958039892 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -2.958039892 + -1.5 y = -2.958039892 + -1.5 Combine like terms: -2.958039892 + -1.5 = -4.458039892 y = -4.458039892 Simplifying y = -4.458039892Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.458039892, -4.458039892}
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