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Simplifying 3y2 + -5y + 5y2 + -7 = 0 Reorder the terms: -7 + -5y + 3y2 + 5y2 = 0 Combine like terms: 3y2 + 5y2 = 8y2 -7 + -5y + 8y2 = 0 Solving -7 + -5y + 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'. -0.875 + -0.625y + y2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + -0.625y + 0.875 + y2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + -0.625y + y2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + -0.625y + y2 = 0 + 0.875 -0.625y + y2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 -0.625y + y2 = 0.875 The y term is -0.625y. Take half its coefficient (-0.3125). Square it (0.09765625) and add it to both sides. Add '0.09765625' to each side of the equation. -0.625y + 0.09765625 + y2 = 0.875 + 0.09765625 Reorder the terms: 0.09765625 + -0.625y + y2 = 0.875 + 0.09765625 Combine like terms: 0.875 + 0.09765625 = 0.97265625 0.09765625 + -0.625y + y2 = 0.97265625 Factor a perfect square on the left side: (y + -0.3125)(y + -0.3125) = 0.97265625 Calculate the square root of the right side: 0.986233365 Break this problem into two subproblems by setting (y + -0.3125) equal to 0.986233365 and -0.986233365.Subproblem 1
y + -0.3125 = 0.986233365 Simplifying y + -0.3125 = 0.986233365 Reorder the terms: -0.3125 + y = 0.986233365 Solving -0.3125 + y = 0.986233365 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + y = 0.986233365 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + y = 0.986233365 + 0.3125 y = 0.986233365 + 0.3125 Combine like terms: 0.986233365 + 0.3125 = 1.298733365 y = 1.298733365 Simplifying y = 1.298733365Subproblem 2
y + -0.3125 = -0.986233365 Simplifying y + -0.3125 = -0.986233365 Reorder the terms: -0.3125 + y = -0.986233365 Solving -0.3125 + y = -0.986233365 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.3125' to each side of the equation. -0.3125 + 0.3125 + y = -0.986233365 + 0.3125 Combine like terms: -0.3125 + 0.3125 = 0.0000 0.0000 + y = -0.986233365 + 0.3125 y = -0.986233365 + 0.3125 Combine like terms: -0.986233365 + 0.3125 = -0.673733365 y = -0.673733365 Simplifying y = -0.673733365Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.298733365, -0.673733365}
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