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Simplifying 3y(2y + -5) + -1(3y + -2)(4y + 4) = 0 Reorder the terms: 3y(-5 + 2y) + -1(3y + -2)(4y + 4) = 0 (-5 * 3y + 2y * 3y) + -1(3y + -2)(4y + 4) = 0 (-15y + 6y2) + -1(3y + -2)(4y + 4) = 0 Reorder the terms: -15y + 6y2 + -1(-2 + 3y)(4y + 4) = 0 Reorder the terms: -15y + 6y2 + -1(-2 + 3y)(4 + 4y) = 0 Multiply (-2 + 3y) * (4 + 4y) -15y + 6y2 + -1(-2(4 + 4y) + 3y * (4 + 4y)) = 0 -15y + 6y2 + -1((4 * -2 + 4y * -2) + 3y * (4 + 4y)) = 0 -15y + 6y2 + -1((-8 + -8y) + 3y * (4 + 4y)) = 0 -15y + 6y2 + -1(-8 + -8y + (4 * 3y + 4y * 3y)) = 0 -15y + 6y2 + -1(-8 + -8y + (12y + 12y2)) = 0 Combine like terms: -8y + 12y = 4y -15y + 6y2 + -1(-8 + 4y + 12y2) = 0 -15y + 6y2 + (-8 * -1 + 4y * -1 + 12y2 * -1) = 0 -15y + 6y2 + (8 + -4y + -12y2) = 0 Reorder the terms: 8 + -15y + -4y + 6y2 + -12y2 = 0 Combine like terms: -15y + -4y = -19y 8 + -19y + 6y2 + -12y2 = 0 Combine like terms: 6y2 + -12y2 = -6y2 8 + -19y + -6y2 = 0 Solving 8 + -19y + -6y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.333333333 + 3.166666667y + y2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 3.166666667y + 1.333333333 + y2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 3.166666667y + y2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 3.166666667y + y2 = 0 + 1.333333333 3.166666667y + y2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 3.166666667y + y2 = 1.333333333 The y term is 3.166666667y. Take half its coefficient (1.583333334). Square it (2.506944447) and add it to both sides. Add '2.506944447' to each side of the equation. 3.166666667y + 2.506944447 + y2 = 1.333333333 + 2.506944447 Reorder the terms: 2.506944447 + 3.166666667y + y2 = 1.333333333 + 2.506944447 Combine like terms: 1.333333333 + 2.506944447 = 3.84027778 2.506944447 + 3.166666667y + y2 = 3.84027778 Factor a perfect square on the left side: (y + 1.583333334)(y + 1.583333334) = 3.84027778 Calculate the square root of the right side: 1.95966267 Break this problem into two subproblems by setting (y + 1.583333334) equal to 1.95966267 and -1.95966267.Subproblem 1
y + 1.583333334 = 1.95966267 Simplifying y + 1.583333334 = 1.95966267 Reorder the terms: 1.583333334 + y = 1.95966267 Solving 1.583333334 + y = 1.95966267 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + y = 1.95966267 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + y = 1.95966267 + -1.583333334 y = 1.95966267 + -1.583333334 Combine like terms: 1.95966267 + -1.583333334 = 0.376329336 y = 0.376329336 Simplifying y = 0.376329336Subproblem 2
y + 1.583333334 = -1.95966267 Simplifying y + 1.583333334 = -1.95966267 Reorder the terms: 1.583333334 + y = -1.95966267 Solving 1.583333334 + y = -1.95966267 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-1.583333334' to each side of the equation. 1.583333334 + -1.583333334 + y = -1.95966267 + -1.583333334 Combine like terms: 1.583333334 + -1.583333334 = 0.000000000 0.000000000 + y = -1.95966267 + -1.583333334 y = -1.95966267 + -1.583333334 Combine like terms: -1.95966267 + -1.583333334 = -3.542996004 y = -3.542996004 Simplifying y = -3.542996004Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.376329336, -3.542996004}
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