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Simplifying (y + -5)(2y + 3) + -8 = 0 Reorder the terms: (-5 + y)(2y + 3) + -8 = 0 Reorder the terms: (-5 + y)(3 + 2y) + -8 = 0 Multiply (-5 + y) * (3 + 2y) (-5(3 + 2y) + y(3 + 2y)) + -8 = 0 ((3 * -5 + 2y * -5) + y(3 + 2y)) + -8 = 0 ((-15 + -10y) + y(3 + 2y)) + -8 = 0 (-15 + -10y + (3 * y + 2y * y)) + -8 = 0 (-15 + -10y + (3y + 2y2)) + -8 = 0 Combine like terms: -10y + 3y = -7y (-15 + -7y + 2y2) + -8 = 0 Reorder the terms: -15 + -8 + -7y + 2y2 = 0 Combine like terms: -15 + -8 = -23 -23 + -7y + 2y2 = 0 Solving -23 + -7y + 2y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11.5 + -3.5y + y2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + -3.5y + 11.5 + y2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + -3.5y + y2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + -3.5y + y2 = 0 + 11.5 -3.5y + y2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 -3.5y + y2 = 11.5 The y term is -3.5y. Take half its coefficient (-1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. -3.5y + 3.0625 + y2 = 11.5 + 3.0625 Reorder the terms: 3.0625 + -3.5y + y2 = 11.5 + 3.0625 Combine like terms: 11.5 + 3.0625 = 14.5625 3.0625 + -3.5y + y2 = 14.5625 Factor a perfect square on the left side: (y + -1.75)(y + -1.75) = 14.5625 Calculate the square root of the right side: 3.816084381 Break this problem into two subproblems by setting (y + -1.75) equal to 3.816084381 and -3.816084381.Subproblem 1
y + -1.75 = 3.816084381 Simplifying y + -1.75 = 3.816084381 Reorder the terms: -1.75 + y = 3.816084381 Solving -1.75 + y = 3.816084381 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = 3.816084381 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = 3.816084381 + 1.75 y = 3.816084381 + 1.75 Combine like terms: 3.816084381 + 1.75 = 5.566084381 y = 5.566084381 Simplifying y = 5.566084381Subproblem 2
y + -1.75 = -3.816084381 Simplifying y + -1.75 = -3.816084381 Reorder the terms: -1.75 + y = -3.816084381 Solving -1.75 + y = -3.816084381 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.75' to each side of the equation. -1.75 + 1.75 + y = -3.816084381 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + y = -3.816084381 + 1.75 y = -3.816084381 + 1.75 Combine like terms: -3.816084381 + 1.75 = -2.066084381 y = -2.066084381 Simplifying y = -2.066084381Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.566084381, -2.066084381}
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