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Simplifying (3y2 + -2y + 1) + (5y2 + -1y + -4) = 0 Reorder the terms: (1 + -2y + 3y2) + (5y2 + -1y + -4) = 0 Remove parenthesis around (1 + -2y + 3y2) 1 + -2y + 3y2 + (5y2 + -1y + -4) = 0 Reorder the terms: 1 + -2y + 3y2 + (-4 + -1y + 5y2) = 0 Remove parenthesis around (-4 + -1y + 5y2) 1 + -2y + 3y2 + -4 + -1y + 5y2 = 0 Reorder the terms: 1 + -4 + -2y + -1y + 3y2 + 5y2 = 0 Combine like terms: 1 + -4 = -3 -3 + -2y + -1y + 3y2 + 5y2 = 0 Combine like terms: -2y + -1y = -3y -3 + -3y + 3y2 + 5y2 = 0 Combine like terms: 3y2 + 5y2 = 8y2 -3 + -3y + 8y2 = 0 Solving -3 + -3y + 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 + -0.375y + y2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + -0.375y + 0.375 + y2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + -0.375y + y2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + -0.375y + y2 = 0 + 0.375 -0.375y + y2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 -0.375y + y2 = 0.375 The y term is -0.375y. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375y + 0.03515625 + y2 = 0.375 + 0.03515625 Reorder the terms: 0.03515625 + -0.375y + y2 = 0.375 + 0.03515625 Combine like terms: 0.375 + 0.03515625 = 0.41015625 0.03515625 + -0.375y + y2 = 0.41015625 Factor a perfect square on the left side: (y + -0.1875)(y + -0.1875) = 0.41015625 Calculate the square root of the right side: 0.640434423 Break this problem into two subproblems by setting (y + -0.1875) equal to 0.640434423 and -0.640434423.Subproblem 1
y + -0.1875 = 0.640434423 Simplifying y + -0.1875 = 0.640434423 Reorder the terms: -0.1875 + y = 0.640434423 Solving -0.1875 + y = 0.640434423 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + y = 0.640434423 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + y = 0.640434423 + 0.1875 y = 0.640434423 + 0.1875 Combine like terms: 0.640434423 + 0.1875 = 0.827934423 y = 0.827934423 Simplifying y = 0.827934423Subproblem 2
y + -0.1875 = -0.640434423 Simplifying y + -0.1875 = -0.640434423 Reorder the terms: -0.1875 + y = -0.640434423 Solving -0.1875 + y = -0.640434423 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + y = -0.640434423 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + y = -0.640434423 + 0.1875 y = -0.640434423 + 0.1875 Combine like terms: -0.640434423 + 0.1875 = -0.452934423 y = -0.452934423 Simplifying y = -0.452934423Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.827934423, -0.452934423}
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