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Simplifying (5y2 + -3y + -7) + (4y2 + -6y + 1) = 0 Reorder the terms: (-7 + -3y + 5y2) + (4y2 + -6y + 1) = 0 Remove parenthesis around (-7 + -3y + 5y2) -7 + -3y + 5y2 + (4y2 + -6y + 1) = 0 Reorder the terms: -7 + -3y + 5y2 + (1 + -6y + 4y2) = 0 Remove parenthesis around (1 + -6y + 4y2) -7 + -3y + 5y2 + 1 + -6y + 4y2 = 0 Reorder the terms: -7 + 1 + -3y + -6y + 5y2 + 4y2 = 0 Combine like terms: -7 + 1 = -6 -6 + -3y + -6y + 5y2 + 4y2 = 0 Combine like terms: -3y + -6y = -9y -6 + -9y + 5y2 + 4y2 = 0 Combine like terms: 5y2 + 4y2 = 9y2 -6 + -9y + 9y2 = 0 Solving -6 + -9y + 9y2 = 0 Solving for variable 'y'. Factor out the Greatest Common Factor (GCF), '3'. 3(-2 + -3y + 3y2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-2 + -3y + 3y2)' equal to zero and attempt to solve: Simplifying -2 + -3y + 3y2 = 0 Solving -2 + -3y + 3y2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.6666666667 + -1y + y2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -1y + 0.6666666667 + y2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -1y + y2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -1y + y2 = 0 + 0.6666666667 -1y + y2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -1y + y2 = 0.6666666667 The y term is -1y. 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. -1y + 0.25 + y2 = 0.6666666667 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 0.6666666667 + 0.25 Combine like terms: 0.6666666667 + 0.25 = 0.9166666667 0.25 + -1y + y2 = 0.9166666667 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 0.9166666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (y + -0.5) equal to 0.957427108 and -0.957427108.Subproblem 1
y + -0.5 = 0.957427108 Simplifying y + -0.5 = 0.957427108 Reorder the terms: -0.5 + y = 0.957427108 Solving -0.5 + y = 0.957427108 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 = 0.957427108 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 0.957427108 + 0.5 y = 0.957427108 + 0.5 Combine like terms: 0.957427108 + 0.5 = 1.457427108 y = 1.457427108 Simplifying y = 1.457427108Subproblem 2
y + -0.5 = -0.957427108 Simplifying y + -0.5 = -0.957427108 Reorder the terms: -0.5 + y = -0.957427108 Solving -0.5 + y = -0.957427108 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 = -0.957427108 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -0.957427108 + 0.5 y = -0.957427108 + 0.5 Combine like terms: -0.957427108 + 0.5 = -0.457427108 y = -0.457427108 Simplifying y = -0.457427108Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.457427108, -0.457427108}Solution
y = {1.457427108, -0.457427108}
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