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Simplifying (y2 + -4y + 3) + -1(4y2 + 5y + -2) = 0 Reorder the terms: (3 + -4y + y2) + -1(4y2 + 5y + -2) = 0 Remove parenthesis around (3 + -4y + y2) 3 + -4y + y2 + -1(4y2 + 5y + -2) = 0 Reorder the terms: 3 + -4y + y2 + -1(-2 + 5y + 4y2) = 0 3 + -4y + y2 + (-2 * -1 + 5y * -1 + 4y2 * -1) = 0 3 + -4y + y2 + (2 + -5y + -4y2) = 0 Reorder the terms: 3 + 2 + -4y + -5y + y2 + -4y2 = 0 Combine like terms: 3 + 2 = 5 5 + -4y + -5y + y2 + -4y2 = 0 Combine like terms: -4y + -5y = -9y 5 + -9y + y2 + -4y2 = 0 Combine like terms: y2 + -4y2 = -3y2 5 + -9y + -3y2 = 0 Solving 5 + -9y + -3y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -1.666666667 + 3y + y2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 3y + 1.666666667 + y2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 3y + y2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 3y + y2 = 0 + 1.666666667 3y + y2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 3y + y2 = 1.666666667 The y term is 3y. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3y + 2.25 + y2 = 1.666666667 + 2.25 Reorder the terms: 2.25 + 3y + y2 = 1.666666667 + 2.25 Combine like terms: 1.666666667 + 2.25 = 3.916666667 2.25 + 3y + y2 = 3.916666667 Factor a perfect square on the left side: (y + 1.5)(y + 1.5) = 3.916666667 Calculate the square root of the right side: 1.979057015 Break this problem into two subproblems by setting (y + 1.5) equal to 1.979057015 and -1.979057015.Subproblem 1
y + 1.5 = 1.979057015 Simplifying y + 1.5 = 1.979057015 Reorder the terms: 1.5 + y = 1.979057015 Solving 1.5 + y = 1.979057015 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = 1.979057015 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = 1.979057015 + -1.5 y = 1.979057015 + -1.5 Combine like terms: 1.979057015 + -1.5 = 0.479057015 y = 0.479057015 Simplifying y = 0.479057015Subproblem 2
y + 1.5 = -1.979057015 Simplifying y + 1.5 = -1.979057015 Reorder the terms: 1.5 + y = -1.979057015 Solving 1.5 + y = -1.979057015 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + y = -1.979057015 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + y = -1.979057015 + -1.5 y = -1.979057015 + -1.5 Combine like terms: -1.979057015 + -1.5 = -3.479057015 y = -3.479057015 Simplifying y = -3.479057015Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.479057015, -3.479057015}
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