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Simplifying 45y2 + -130y + 84 = 0 Reorder the terms: 84 + -130y + 45y2 = 0 Solving 84 + -130y + 45y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. 1.866666667 + -2.888888889y + y2 = 0 Move the constant term to the right: Add '-1.866666667' to each side of the equation. 1.866666667 + -2.888888889y + -1.866666667 + y2 = 0 + -1.866666667 Reorder the terms: 1.866666667 + -1.866666667 + -2.888888889y + y2 = 0 + -1.866666667 Combine like terms: 1.866666667 + -1.866666667 = 0.000000000 0.000000000 + -2.888888889y + y2 = 0 + -1.866666667 -2.888888889y + y2 = 0 + -1.866666667 Combine like terms: 0 + -1.866666667 = -1.866666667 -2.888888889y + y2 = -1.866666667 The y term is -2.888888889y. Take half its coefficient (-1.444444445). Square it (2.086419755) and add it to both sides. Add '2.086419755' to each side of the equation. -2.888888889y + 2.086419755 + y2 = -1.866666667 + 2.086419755 Reorder the terms: 2.086419755 + -2.888888889y + y2 = -1.866666667 + 2.086419755 Combine like terms: -1.866666667 + 2.086419755 = 0.219753088 2.086419755 + -2.888888889y + y2 = 0.219753088 Factor a perfect square on the left side: (y + -1.444444445)(y + -1.444444445) = 0.219753088 Calculate the square root of the right side: 0.468778293 Break this problem into two subproblems by setting (y + -1.444444445) equal to 0.468778293 and -0.468778293.Subproblem 1
y + -1.444444445 = 0.468778293 Simplifying y + -1.444444445 = 0.468778293 Reorder the terms: -1.444444445 + y = 0.468778293 Solving -1.444444445 + y = 0.468778293 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + y = 0.468778293 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + y = 0.468778293 + 1.444444445 y = 0.468778293 + 1.444444445 Combine like terms: 0.468778293 + 1.444444445 = 1.913222738 y = 1.913222738 Simplifying y = 1.913222738Subproblem 2
y + -1.444444445 = -0.468778293 Simplifying y + -1.444444445 = -0.468778293 Reorder the terms: -1.444444445 + y = -0.468778293 Solving -1.444444445 + y = -0.468778293 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.444444445' to each side of the equation. -1.444444445 + 1.444444445 + y = -0.468778293 + 1.444444445 Combine like terms: -1.444444445 + 1.444444445 = 0.000000000 0.000000000 + y = -0.468778293 + 1.444444445 y = -0.468778293 + 1.444444445 Combine like terms: -0.468778293 + 1.444444445 = 0.975666152 y = 0.975666152 Simplifying y = 0.975666152Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.913222738, 0.975666152}
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