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Simplifying (4y2 + -5y) + (3y + -7y2) + -1(2y2 + 6y + -5) = 0 Reorder the terms: (-5y + 4y2) + (3y + -7y2) + -1(2y2 + 6y + -5) = 0 Remove parenthesis around (-5y + 4y2) -5y + 4y2 + (3y + -7y2) + -1(2y2 + 6y + -5) = 0 Remove parenthesis around (3y + -7y2) -5y + 4y2 + 3y + -7y2 + -1(2y2 + 6y + -5) = 0 Reorder the terms: -5y + 4y2 + 3y + -7y2 + -1(-5 + 6y + 2y2) = 0 -5y + 4y2 + 3y + -7y2 + (-5 * -1 + 6y * -1 + 2y2 * -1) = 0 -5y + 4y2 + 3y + -7y2 + (5 + -6y + -2y2) = 0 Reorder the terms: 5 + -5y + 3y + -6y + 4y2 + -7y2 + -2y2 = 0 Combine like terms: -5y + 3y = -2y 5 + -2y + -6y + 4y2 + -7y2 + -2y2 = 0 Combine like terms: -2y + -6y = -8y 5 + -8y + 4y2 + -7y2 + -2y2 = 0 Combine like terms: 4y2 + -7y2 = -3y2 5 + -8y + -3y2 + -2y2 = 0 Combine like terms: -3y2 + -2y2 = -5y2 5 + -8y + -5y2 = 0 Solving 5 + -8y + -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 + 1.6y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.6y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + 1.6y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.6y + y2 = 0 + 1 1.6y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.6y + y2 = 1 The y term is 1.6y. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6y + 0.64 + y2 = 1 + 0.64 Reorder the terms: 0.64 + 1.6y + y2 = 1 + 0.64 Combine like terms: 1 + 0.64 = 1.64 0.64 + 1.6y + y2 = 1.64 Factor a perfect square on the left side: (y + 0.8)(y + 0.8) = 1.64 Calculate the square root of the right side: 1.280624847 Break this problem into two subproblems by setting (y + 0.8) equal to 1.280624847 and -1.280624847.Subproblem 1
y + 0.8 = 1.280624847 Simplifying y + 0.8 = 1.280624847 Reorder the terms: 0.8 + y = 1.280624847 Solving 0.8 + y = 1.280624847 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + y = 1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + y = 1.280624847 + -0.8 y = 1.280624847 + -0.8 Combine like terms: 1.280624847 + -0.8 = 0.480624847 y = 0.480624847 Simplifying y = 0.480624847Subproblem 2
y + 0.8 = -1.280624847 Simplifying y + 0.8 = -1.280624847 Reorder the terms: 0.8 + y = -1.280624847 Solving 0.8 + y = -1.280624847 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + y = -1.280624847 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + y = -1.280624847 + -0.8 y = -1.280624847 + -0.8 Combine like terms: -1.280624847 + -0.8 = -2.080624847 y = -2.080624847 Simplifying y = -2.080624847Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.480624847, -2.080624847}
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