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Simplifying 32y(12 + -5y) = 8 + 7y (12 * 32y + -5y * 32y) = 8 + 7y (384y + -160y2) = 8 + 7y Solving 384y + -160y2 = 8 + 7y Solving for variable 'y'. Reorder the terms: -8 + 384y + -7y + -160y2 = 8 + 7y + -8 + -7y Combine like terms: 384y + -7y = 377y -8 + 377y + -160y2 = 8 + 7y + -8 + -7y Reorder the terms: -8 + 377y + -160y2 = 8 + -8 + 7y + -7y Combine like terms: 8 + -8 = 0 -8 + 377y + -160y2 = 0 + 7y + -7y -8 + 377y + -160y2 = 7y + -7y Combine like terms: 7y + -7y = 0 -8 + 377y + -160y2 = 0 Begin completing the square. Divide all terms by -160 the coefficient of the squared term: Divide each side by '-160'. 0.05 + -2.35625y + y2 = 0.0 Move the constant term to the right: Add '-0.05' to each side of the equation. 0.05 + -2.35625y + -0.05 + y2 = 0.0 + -0.05 Reorder the terms: 0.05 + -0.05 + -2.35625y + y2 = 0.0 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + -2.35625y + y2 = 0.0 + -0.05 -2.35625y + y2 = 0.0 + -0.05 Combine like terms: 0.0 + -0.05 = -0.05 -2.35625y + y2 = -0.05 The y term is -2.35625y. Take half its coefficient (-1.178125). Square it (1.387978516) and add it to both sides. Add '1.387978516' to each side of the equation. -2.35625y + 1.387978516 + y2 = -0.05 + 1.387978516 Reorder the terms: 1.387978516 + -2.35625y + y2 = -0.05 + 1.387978516 Combine like terms: -0.05 + 1.387978516 = 1.337978516 1.387978516 + -2.35625y + y2 = 1.337978516 Factor a perfect square on the left side: (y + -1.178125)(y + -1.178125) = 1.337978516 Calculate the square root of the right side: 1.156710213 Break this problem into two subproblems by setting (y + -1.178125) equal to 1.156710213 and -1.156710213.Subproblem 1
y + -1.178125 = 1.156710213 Simplifying y + -1.178125 = 1.156710213 Reorder the terms: -1.178125 + y = 1.156710213 Solving -1.178125 + y = 1.156710213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.178125' to each side of the equation. -1.178125 + 1.178125 + y = 1.156710213 + 1.178125 Combine like terms: -1.178125 + 1.178125 = 0.000000 0.000000 + y = 1.156710213 + 1.178125 y = 1.156710213 + 1.178125 Combine like terms: 1.156710213 + 1.178125 = 2.334835213 y = 2.334835213 Simplifying y = 2.334835213Subproblem 2
y + -1.178125 = -1.156710213 Simplifying y + -1.178125 = -1.156710213 Reorder the terms: -1.178125 + y = -1.156710213 Solving -1.178125 + y = -1.156710213 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.178125' to each side of the equation. -1.178125 + 1.178125 + y = -1.156710213 + 1.178125 Combine like terms: -1.178125 + 1.178125 = 0.000000 0.000000 + y = -1.156710213 + 1.178125 y = -1.156710213 + 1.178125 Combine like terms: -1.156710213 + 1.178125 = 0.021414787 y = 0.021414787 Simplifying y = 0.021414787Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.334835213, 0.021414787}
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