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Simplifying (-5y2 + -3y + 3) + -1(5y2 + -3y + -8) + (6y2 + 6y + -9) = 0 Reorder the terms: (3 + -3y + -5y2) + -1(5y2 + -3y + -8) + (6y2 + 6y + -9) = 0 Remove parenthesis around (3 + -3y + -5y2) 3 + -3y + -5y2 + -1(5y2 + -3y + -8) + (6y2 + 6y + -9) = 0 Reorder the terms: 3 + -3y + -5y2 + -1(-8 + -3y + 5y2) + (6y2 + 6y + -9) = 0 3 + -3y + -5y2 + (-8 * -1 + -3y * -1 + 5y2 * -1) + (6y2 + 6y + -9) = 0 3 + -3y + -5y2 + (8 + 3y + -5y2) + (6y2 + 6y + -9) = 0 Reorder the terms: 3 + -3y + -5y2 + 8 + 3y + -5y2 + (-9 + 6y + 6y2) = 0 Remove parenthesis around (-9 + 6y + 6y2) 3 + -3y + -5y2 + 8 + 3y + -5y2 + -9 + 6y + 6y2 = 0 Reorder the terms: 3 + 8 + -9 + -3y + 3y + 6y + -5y2 + -5y2 + 6y2 = 0 Combine like terms: 3 + 8 = 11 11 + -9 + -3y + 3y + 6y + -5y2 + -5y2 + 6y2 = 0 Combine like terms: 11 + -9 = 2 2 + -3y + 3y + 6y + -5y2 + -5y2 + 6y2 = 0 Combine like terms: -3y + 3y = 0 2 + 0 + 6y + -5y2 + -5y2 + 6y2 = 0 2 + 6y + -5y2 + -5y2 + 6y2 = 0 Combine like terms: -5y2 + -5y2 = -10y2 2 + 6y + -10y2 + 6y2 = 0 Combine like terms: -10y2 + 6y2 = -4y2 2 + 6y + -4y2 = 0 Solving 2 + 6y + -4y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 3y + -2y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 3y + -2y2)' equal to zero and attempt to solve: Simplifying 1 + 3y + -2y2 = 0 Solving 1 + 3y + -2y2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -1.5y + y2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -1.5y + 0.5 + y2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -1.5y + y2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -1.5y + y2 = 0 + 0.5 -1.5y + y2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -1.5y + y2 = 0.5 The y term is -1.5y. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5y + 0.5625 + y2 = 0.5 + 0.5625 Reorder the terms: 0.5625 + -1.5y + y2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + -1.5y + y2 = 1.0625 Factor a perfect square on the left side: (y + -0.75)(y + -0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (y + -0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
y + -0.75 = 1.030776406 Simplifying y + -0.75 = 1.030776406 Reorder the terms: -0.75 + y = 1.030776406 Solving -0.75 + y = 1.030776406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = 1.030776406 + 0.75 y = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 y = 1.780776406 Simplifying y = 1.780776406Subproblem 2
y + -0.75 = -1.030776406 Simplifying y + -0.75 = -1.030776406 Reorder the terms: -0.75 + y = -1.030776406 Solving -0.75 + y = -1.030776406 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + y = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + y = -1.030776406 + 0.75 y = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 y = -0.280776406 Simplifying y = -0.280776406Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.780776406, -0.280776406}Solution
y = {1.780776406, -0.280776406}
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