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Simplifying (4y + -5)(2y + -5) = 105 Reorder the terms: (-5 + 4y)(2y + -5) = 105 Reorder the terms: (-5 + 4y)(-5 + 2y) = 105 Multiply (-5 + 4y) * (-5 + 2y) (-5(-5 + 2y) + 4y * (-5 + 2y)) = 105 ((-5 * -5 + 2y * -5) + 4y * (-5 + 2y)) = 105 ((25 + -10y) + 4y * (-5 + 2y)) = 105 (25 + -10y + (-5 * 4y + 2y * 4y)) = 105 (25 + -10y + (-20y + 8y2)) = 105 Combine like terms: -10y + -20y = -30y (25 + -30y + 8y2) = 105 Solving 25 + -30y + 8y2 = 105 Solving for variable 'y'. Reorder the terms: 25 + -105 + -30y + 8y2 = 105 + -105 Combine like terms: 25 + -105 = -80 -80 + -30y + 8y2 = 105 + -105 Combine like terms: 105 + -105 = 0 -80 + -30y + 8y2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-40 + -15y + 4y2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-40 + -15y + 4y2)' equal to zero and attempt to solve: Simplifying -40 + -15y + 4y2 = 0 Solving -40 + -15y + 4y2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -10 + -3.75y + y2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -3.75y + 10 + y2 = 0 + 10 Reorder the terms: -10 + 10 + -3.75y + y2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -3.75y + y2 = 0 + 10 -3.75y + y2 = 0 + 10 Combine like terms: 0 + 10 = 10 -3.75y + y2 = 10 The y term is -3.75y. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75y + 3.515625 + y2 = 10 + 3.515625 Reorder the terms: 3.515625 + -3.75y + y2 = 10 + 3.515625 Combine like terms: 10 + 3.515625 = 13.515625 3.515625 + -3.75y + y2 = 13.515625 Factor a perfect square on the left side: (y + -1.875)(y + -1.875) = 13.515625 Calculate the square root of the right side: 3.676360292 Break this problem into two subproblems by setting (y + -1.875) equal to 3.676360292 and -3.676360292.Subproblem 1
y + -1.875 = 3.676360292 Simplifying y + -1.875 = 3.676360292 Reorder the terms: -1.875 + y = 3.676360292 Solving -1.875 + y = 3.676360292 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = 3.676360292 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = 3.676360292 + 1.875 y = 3.676360292 + 1.875 Combine like terms: 3.676360292 + 1.875 = 5.551360292 y = 5.551360292 Simplifying y = 5.551360292Subproblem 2
y + -1.875 = -3.676360292 Simplifying y + -1.875 = -3.676360292 Reorder the terms: -1.875 + y = -3.676360292 Solving -1.875 + y = -3.676360292 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + y = -3.676360292 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + y = -3.676360292 + 1.875 y = -3.676360292 + 1.875 Combine like terms: -3.676360292 + 1.875 = -1.801360292 y = -1.801360292 Simplifying y = -1.801360292Solution
The solution to the problem is based on the solutions from the subproblems. y = {5.551360292, -1.801360292}Solution
y = {5.551360292, -1.801360292}
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