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Simplifying 0 = 4y2 + 33y + 23 Reorder the terms: 0 = 23 + 33y + 4y2 Solving 0 = 23 + 33y + 4y2 Solving for variable 'y'. Combine like terms: 0 + -23 = -23 -23 + -33y + -4y2 = 23 + 33y + 4y2 + -23 + -33y + -4y2 Reorder the terms: -23 + -33y + -4y2 = 23 + -23 + 33y + -33y + 4y2 + -4y2 Combine like terms: 23 + -23 = 0 -23 + -33y + -4y2 = 0 + 33y + -33y + 4y2 + -4y2 -23 + -33y + -4y2 = 33y + -33y + 4y2 + -4y2 Combine like terms: 33y + -33y = 0 -23 + -33y + -4y2 = 0 + 4y2 + -4y2 -23 + -33y + -4y2 = 4y2 + -4y2 Combine like terms: 4y2 + -4y2 = 0 -23 + -33y + -4y2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(23 + 33y + 4y2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(23 + 33y + 4y2)' equal to zero and attempt to solve: Simplifying 23 + 33y + 4y2 = 0 Solving 23 + 33y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.75 + 8.25y + y2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + 8.25y + -5.75 + y2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + 8.25y + y2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + 8.25y + y2 = 0 + -5.75 8.25y + y2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 8.25y + y2 = -5.75 The y term is 8.25y. Take half its coefficient (4.125). Square it (17.015625) and add it to both sides. Add '17.015625' to each side of the equation. 8.25y + 17.015625 + y2 = -5.75 + 17.015625 Reorder the terms: 17.015625 + 8.25y + y2 = -5.75 + 17.015625 Combine like terms: -5.75 + 17.015625 = 11.265625 17.015625 + 8.25y + y2 = 11.265625 Factor a perfect square on the left side: (y + 4.125)(y + 4.125) = 11.265625 Calculate the square root of the right side: 3.356430396 Break this problem into two subproblems by setting (y + 4.125) equal to 3.356430396 and -3.356430396.Subproblem 1
y + 4.125 = 3.356430396 Simplifying y + 4.125 = 3.356430396 Reorder the terms: 4.125 + y = 3.356430396 Solving 4.125 + y = 3.356430396 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4.125' to each side of the equation. 4.125 + -4.125 + y = 3.356430396 + -4.125 Combine like terms: 4.125 + -4.125 = 0.000 0.000 + y = 3.356430396 + -4.125 y = 3.356430396 + -4.125 Combine like terms: 3.356430396 + -4.125 = -0.768569604 y = -0.768569604 Simplifying y = -0.768569604Subproblem 2
y + 4.125 = -3.356430396 Simplifying y + 4.125 = -3.356430396 Reorder the terms: 4.125 + y = -3.356430396 Solving 4.125 + y = -3.356430396 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-4.125' to each side of the equation. 4.125 + -4.125 + y = -3.356430396 + -4.125 Combine like terms: 4.125 + -4.125 = 0.000 0.000 + y = -3.356430396 + -4.125 y = -3.356430396 + -4.125 Combine like terms: -3.356430396 + -4.125 = -7.481430396 y = -7.481430396 Simplifying y = -7.481430396Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.768569604, -7.481430396}Solution
y = {-0.768569604, -7.481430396}
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