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Simplifying (3 + -1y) * y = 5y2 + -4 Reorder the terms for easier multiplication: y(3 + -1y) = 5y2 + -4 (3 * y + -1y * y) = 5y2 + -4 (3y + -1y2) = 5y2 + -4 Reorder the terms: 3y + -1y2 = -4 + 5y2 Solving 3y + -1y2 = -4 + 5y2 Solving for variable 'y'. Reorder the terms: 4 + 3y + -1y2 + -5y2 = -4 + 5y2 + 4 + -5y2 Combine like terms: -1y2 + -5y2 = -6y2 4 + 3y + -6y2 = -4 + 5y2 + 4 + -5y2 Reorder the terms: 4 + 3y + -6y2 = -4 + 4 + 5y2 + -5y2 Combine like terms: -4 + 4 = 0 4 + 3y + -6y2 = 0 + 5y2 + -5y2 4 + 3y + -6y2 = 5y2 + -5y2 Combine like terms: 5y2 + -5y2 = 0 4 + 3y + -6y2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -0.6666666667 + -0.5y + y2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.5y + 0.6666666667 + y2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.5y + y2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.5y + y2 = 0 + 0.6666666667 -0.5y + y2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.5y + y2 = 0.6666666667 The y term is -0.5y. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5y + 0.0625 + y2 = 0.6666666667 + 0.0625 Reorder the terms: 0.0625 + -0.5y + y2 = 0.6666666667 + 0.0625 Combine like terms: 0.6666666667 + 0.0625 = 0.7291666667 0.0625 + -0.5y + y2 = 0.7291666667 Factor a perfect square on the left side: (y + -0.25)(y + -0.25) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (y + -0.25) equal to 0.853912564 and -0.853912564.Subproblem 1
y + -0.25 = 0.853912564 Simplifying y + -0.25 = 0.853912564 Reorder the terms: -0.25 + y = 0.853912564 Solving -0.25 + y = 0.853912564 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = 0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = 0.853912564 + 0.25 y = 0.853912564 + 0.25 Combine like terms: 0.853912564 + 0.25 = 1.103912564 y = 1.103912564 Simplifying y = 1.103912564Subproblem 2
y + -0.25 = -0.853912564 Simplifying y + -0.25 = -0.853912564 Reorder the terms: -0.25 + y = -0.853912564 Solving -0.25 + y = -0.853912564 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + y = -0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + y = -0.853912564 + 0.25 y = -0.853912564 + 0.25 Combine like terms: -0.853912564 + 0.25 = -0.603912564 y = -0.603912564 Simplifying y = -0.603912564Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.103912564, -0.603912564}
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