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Simplifying 3(5y + 4) = 5y(2y + -3) + 22 Reorder the terms: 3(4 + 5y) = 5y(2y + -3) + 22 (4 * 3 + 5y * 3) = 5y(2y + -3) + 22 (12 + 15y) = 5y(2y + -3) + 22 Reorder the terms: 12 + 15y = 5y(-3 + 2y) + 22 12 + 15y = (-3 * 5y + 2y * 5y) + 22 12 + 15y = (-15y + 10y2) + 22 Reorder the terms: 12 + 15y = 22 + -15y + 10y2 Solving 12 + 15y = 22 + -15y + 10y2 Solving for variable 'y'. Reorder the terms: 12 + -22 + 15y + 15y + -10y2 = 22 + -15y + 10y2 + -22 + 15y + -10y2 Combine like terms: 12 + -22 = -10 -10 + 15y + 15y + -10y2 = 22 + -15y + 10y2 + -22 + 15y + -10y2 Combine like terms: 15y + 15y = 30y -10 + 30y + -10y2 = 22 + -15y + 10y2 + -22 + 15y + -10y2 Reorder the terms: -10 + 30y + -10y2 = 22 + -22 + -15y + 15y + 10y2 + -10y2 Combine like terms: 22 + -22 = 0 -10 + 30y + -10y2 = 0 + -15y + 15y + 10y2 + -10y2 -10 + 30y + -10y2 = -15y + 15y + 10y2 + -10y2 Combine like terms: -15y + 15y = 0 -10 + 30y + -10y2 = 0 + 10y2 + -10y2 -10 + 30y + -10y2 = 10y2 + -10y2 Combine like terms: 10y2 + -10y2 = 0 -10 + 30y + -10y2 = 0 Factor out the Greatest Common Factor (GCF), '10'. 10(-1 + 3y + -1y2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-1 + 3y + -1y2)' equal to zero and attempt to solve: Simplifying -1 + 3y + -1y2 = 0 Solving -1 + 3y + -1y2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 1 + -3y + y2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -3y + -1 + y2 = 0 + -1 Reorder the terms: 1 + -1 + -3y + y2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -3y + y2 = 0 + -1 -3y + y2 = 0 + -1 Combine like terms: 0 + -1 = -1 -3y + y2 = -1 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 + 2.25 Reorder the terms: 2.25 + -3y + y2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + -3y + y2 = 1.25 Factor a perfect square on the left side: (y + -1.5)(y + -1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (y + -1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
y + -1.5 = 1.118033989 Simplifying y + -1.5 = 1.118033989 Reorder the terms: -1.5 + y = 1.118033989 Solving -1.5 + y = 1.118033989 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 = 1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = 1.118033989 + 1.5 y = 1.118033989 + 1.5 Combine like terms: 1.118033989 + 1.5 = 2.618033989 y = 2.618033989 Simplifying y = 2.618033989Subproblem 2
y + -1.5 = -1.118033989 Simplifying y + -1.5 = -1.118033989 Reorder the terms: -1.5 + y = -1.118033989 Solving -1.5 + y = -1.118033989 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 = -1.118033989 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + y = -1.118033989 + 1.5 y = -1.118033989 + 1.5 Combine like terms: -1.118033989 + 1.5 = 0.381966011 y = 0.381966011 Simplifying y = 0.381966011Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.618033989, 0.381966011}Solution
y = {2.618033989, 0.381966011}
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