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Simplifying (5y2 + 4y) + -1(-7y2 + -8y + 7) = 0 Reorder the terms: (4y + 5y2) + -1(-7y2 + -8y + 7) = 0 Remove parenthesis around (4y + 5y2) 4y + 5y2 + -1(-7y2 + -8y + 7) = 0 Reorder the terms: 4y + 5y2 + -1(7 + -8y + -7y2) = 0 4y + 5y2 + (7 * -1 + -8y * -1 + -7y2 * -1) = 0 4y + 5y2 + (-7 + 8y + 7y2) = 0 Reorder the terms: -7 + 4y + 8y + 5y2 + 7y2 = 0 Combine like terms: 4y + 8y = 12y -7 + 12y + 5y2 + 7y2 = 0 Combine like terms: 5y2 + 7y2 = 12y2 -7 + 12y + 12y2 = 0 Solving -7 + 12y + 12y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.5833333333 + y + y2 = 0 Move the constant term to the right: Add '0.5833333333' to each side of the equation. -0.5833333333 + y + 0.5833333333 + y2 = 0 + 0.5833333333 Reorder the terms: -0.5833333333 + 0.5833333333 + y + y2 = 0 + 0.5833333333 Combine like terms: -0.5833333333 + 0.5833333333 = 0.0000000000 0.0000000000 + y + y2 = 0 + 0.5833333333 y + y2 = 0 + 0.5833333333 Combine like terms: 0 + 0.5833333333 = 0.5833333333 y + y2 = 0.5833333333 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + y2 = 0.5833333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 0.5833333333 + 0.25 Combine like terms: 0.5833333333 + 0.25 = 0.8333333333 1.25 + y2 = 0.8333333333 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 0.8333333333 Calculate the square root of the right side: 0.912870929 Break this problem into two subproblems by setting (y + 0.5) equal to 0.912870929 and -0.912870929.Subproblem 1
y + 0.5 = 0.912870929 Simplifying y + 0.5 = 0.912870929 Reorder the terms: 0.5 + y = 0.912870929 Solving 0.5 + y = 0.912870929 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 0.912870929 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 0.912870929 + -0.5 y = 0.912870929 + -0.5 Combine like terms: 0.912870929 + -0.5 = 0.412870929 y = 0.412870929 Simplifying y = 0.412870929Subproblem 2
y + 0.5 = -0.912870929 Simplifying y + 0.5 = -0.912870929 Reorder the terms: 0.5 + y = -0.912870929 Solving 0.5 + y = -0.912870929 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -0.912870929 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -0.912870929 + -0.5 y = -0.912870929 + -0.5 Combine like terms: -0.912870929 + -0.5 = -1.412870929 y = -1.412870929 Simplifying y = -1.412870929Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.412870929, -1.412870929}
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