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Simplifying (3y2 + 5y + -1) + (5y2 + -2y + 2) + (-3y2 + y + -7) = 0 Reorder the terms: (-1 + 5y + 3y2) + (5y2 + -2y + 2) + (-3y2 + y + -7) = 0 Remove parenthesis around (-1 + 5y + 3y2) -1 + 5y + 3y2 + (5y2 + -2y + 2) + (-3y2 + y + -7) = 0 Reorder the terms: -1 + 5y + 3y2 + (2 + -2y + 5y2) + (-3y2 + y + -7) = 0 Remove parenthesis around (2 + -2y + 5y2) -1 + 5y + 3y2 + 2 + -2y + 5y2 + (-3y2 + y + -7) = 0 Reorder the terms: -1 + 5y + 3y2 + 2 + -2y + 5y2 + (-7 + y + -3y2) = 0 Remove parenthesis around (-7 + y + -3y2) -1 + 5y + 3y2 + 2 + -2y + 5y2 + -7 + y + -3y2 = 0 Reorder the terms: -1 + 2 + -7 + 5y + -2y + y + 3y2 + 5y2 + -3y2 = 0 Combine like terms: -1 + 2 = 1 1 + -7 + 5y + -2y + y + 3y2 + 5y2 + -3y2 = 0 Combine like terms: 1 + -7 = -6 -6 + 5y + -2y + y + 3y2 + 5y2 + -3y2 = 0 Combine like terms: 5y + -2y = 3y -6 + 3y + y + 3y2 + 5y2 + -3y2 = 0 Combine like terms: 3y + y = 4y -6 + 4y + 3y2 + 5y2 + -3y2 = 0 Combine like terms: 3y2 + 5y2 = 8y2 -6 + 4y + 8y2 + -3y2 = 0 Combine like terms: 8y2 + -3y2 = 5y2 -6 + 4y + 5y2 = 0 Solving -6 + 4y + 5y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1.2 + 0.8y + y2 = 0 Move the constant term to the right: Add '1.2' to each side of the equation. -1.2 + 0.8y + 1.2 + y2 = 0 + 1.2 Reorder the terms: -1.2 + 1.2 + 0.8y + y2 = 0 + 1.2 Combine like terms: -1.2 + 1.2 = 0.0 0.0 + 0.8y + y2 = 0 + 1.2 0.8y + y2 = 0 + 1.2 Combine like terms: 0 + 1.2 = 1.2 0.8y + y2 = 1.2 The y term is 0.8y. Take half its coefficient (0.4). Square it (0.16) and add it to both sides. Add '0.16' to each side of the equation. 0.8y + 0.16 + y2 = 1.2 + 0.16 Reorder the terms: 0.16 + 0.8y + y2 = 1.2 + 0.16 Combine like terms: 1.2 + 0.16 = 1.36 0.16 + 0.8y + y2 = 1.36 Factor a perfect square on the left side: (y + 0.4)(y + 0.4) = 1.36 Calculate the square root of the right side: 1.166190379 Break this problem into two subproblems by setting (y + 0.4) equal to 1.166190379 and -1.166190379.Subproblem 1
y + 0.4 = 1.166190379 Simplifying y + 0.4 = 1.166190379 Reorder the terms: 0.4 + y = 1.166190379 Solving 0.4 + y = 1.166190379 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + y = 1.166190379 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + y = 1.166190379 + -0.4 y = 1.166190379 + -0.4 Combine like terms: 1.166190379 + -0.4 = 0.766190379 y = 0.766190379 Simplifying y = 0.766190379Subproblem 2
y + 0.4 = -1.166190379 Simplifying y + 0.4 = -1.166190379 Reorder the terms: 0.4 + y = -1.166190379 Solving 0.4 + y = -1.166190379 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.4' to each side of the equation. 0.4 + -0.4 + y = -1.166190379 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + y = -1.166190379 + -0.4 y = -1.166190379 + -0.4 Combine like terms: -1.166190379 + -0.4 = -1.566190379 y = -1.566190379 Simplifying y = -1.566190379Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.766190379, -1.566190379}
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