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Simplifying 5(2y + 3)(5y + -2) = 84 Reorder the terms: 5(3 + 2y)(5y + -2) = 84 Reorder the terms: 5(3 + 2y)(-2 + 5y) = 84 Multiply (3 + 2y) * (-2 + 5y) 5(3(-2 + 5y) + 2y * (-2 + 5y)) = 84 5((-2 * 3 + 5y * 3) + 2y * (-2 + 5y)) = 84 5((-6 + 15y) + 2y * (-2 + 5y)) = 84 5(-6 + 15y + (-2 * 2y + 5y * 2y)) = 84 5(-6 + 15y + (-4y + 10y2)) = 84 Combine like terms: 15y + -4y = 11y 5(-6 + 11y + 10y2) = 84 (-6 * 5 + 11y * 5 + 10y2 * 5) = 84 (-30 + 55y + 50y2) = 84 Solving -30 + 55y + 50y2 = 84 Solving for variable 'y'. Reorder the terms: -30 + -84 + 55y + 50y2 = 84 + -84 Combine like terms: -30 + -84 = -114 -114 + 55y + 50y2 = 84 + -84 Combine like terms: 84 + -84 = 0 -114 + 55y + 50y2 = 0 Begin completing the square. Divide all terms by 50 the coefficient of the squared term: Divide each side by '50'. -2.28 + 1.1y + y2 = 0.0 Move the constant term to the right: Add '2.28' to each side of the equation. -2.28 + 1.1y + 2.28 + y2 = 0.0 + 2.28 Reorder the terms: -2.28 + 2.28 + 1.1y + y2 = 0.0 + 2.28 Combine like terms: -2.28 + 2.28 = 0.00 0.00 + 1.1y + y2 = 0.0 + 2.28 1.1y + y2 = 0.0 + 2.28 Combine like terms: 0.0 + 2.28 = 2.28 1.1y + y2 = 2.28 The y term is 1.1y. Take half its coefficient (0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. 1.1y + 0.3025 + y2 = 2.28 + 0.3025 Reorder the terms: 0.3025 + 1.1y + y2 = 2.28 + 0.3025 Combine like terms: 2.28 + 0.3025 = 2.5825 0.3025 + 1.1y + y2 = 2.5825 Factor a perfect square on the left side: (y + 0.55)(y + 0.55) = 2.5825 Calculate the square root of the right side: 1.607015868 Break this problem into two subproblems by setting (y + 0.55) equal to 1.607015868 and -1.607015868.Subproblem 1
y + 0.55 = 1.607015868 Simplifying y + 0.55 = 1.607015868 Reorder the terms: 0.55 + y = 1.607015868 Solving 0.55 + y = 1.607015868 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + y = 1.607015868 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + y = 1.607015868 + -0.55 y = 1.607015868 + -0.55 Combine like terms: 1.607015868 + -0.55 = 1.057015868 y = 1.057015868 Simplifying y = 1.057015868Subproblem 2
y + 0.55 = -1.607015868 Simplifying y + 0.55 = -1.607015868 Reorder the terms: 0.55 + y = -1.607015868 Solving 0.55 + y = -1.607015868 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + y = -1.607015868 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + y = -1.607015868 + -0.55 y = -1.607015868 + -0.55 Combine like terms: -1.607015868 + -0.55 = -2.157015868 y = -2.157015868 Simplifying y = -2.157015868Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.057015868, -2.157015868}
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