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Simplifying (5y2 + -2y + -4) + (3y2 + -7y + 1) = 0 Reorder the terms: (-4 + -2y + 5y2) + (3y2 + -7y + 1) = 0 Remove parenthesis around (-4 + -2y + 5y2) -4 + -2y + 5y2 + (3y2 + -7y + 1) = 0 Reorder the terms: -4 + -2y + 5y2 + (1 + -7y + 3y2) = 0 Remove parenthesis around (1 + -7y + 3y2) -4 + -2y + 5y2 + 1 + -7y + 3y2 = 0 Reorder the terms: -4 + 1 + -2y + -7y + 5y2 + 3y2 = 0 Combine like terms: -4 + 1 = -3 -3 + -2y + -7y + 5y2 + 3y2 = 0 Combine like terms: -2y + -7y = -9y -3 + -9y + 5y2 + 3y2 = 0 Combine like terms: 5y2 + 3y2 = 8y2 -3 + -9y + 8y2 = 0 Solving -3 + -9y + 8y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + -1.125y + y2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -1.125y + 0.375 + y2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -1.125y + y2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -1.125y + y2 = 0 + 0.375 -1.125y + y2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -1.125y + y2 = 0.375 The y term is -1.125y. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125y + 0.31640625 + y2 = 0.375 + 0.31640625 Reorder the terms: 0.31640625 + -1.125y + y2 = 0.375 + 0.31640625 Combine like terms: 0.375 + 0.31640625 = 0.69140625 0.31640625 + -1.125y + y2 = 0.69140625 Factor a perfect square on the left side: (y + -0.5625)(y + -0.5625) = 0.69140625 Calculate the square root of the right side: 0.831508418 Break this problem into two subproblems by setting (y + -0.5625) equal to 0.831508418 and -0.831508418.Subproblem 1
y + -0.5625 = 0.831508418 Simplifying y + -0.5625 = 0.831508418 Reorder the terms: -0.5625 + y = 0.831508418 Solving -0.5625 + y = 0.831508418 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + y = 0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + y = 0.831508418 + 0.5625 y = 0.831508418 + 0.5625 Combine like terms: 0.831508418 + 0.5625 = 1.394008418 y = 1.394008418 Simplifying y = 1.394008418Subproblem 2
y + -0.5625 = -0.831508418 Simplifying y + -0.5625 = -0.831508418 Reorder the terms: -0.5625 + y = -0.831508418 Solving -0.5625 + y = -0.831508418 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + y = -0.831508418 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + y = -0.831508418 + 0.5625 y = -0.831508418 + 0.5625 Combine like terms: -0.831508418 + 0.5625 = -0.269008418 y = -0.269008418 Simplifying y = -0.269008418Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.394008418, -0.269008418}
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