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Simplifying 5y2 + -13y + -5 = -15y Reorder the terms: -5 + -13y + 5y2 = -15y Solving -5 + -13y + 5y2 = -15y Solving for variable 'y'. Reorder the terms: -5 + -13y + 15y + 5y2 = -15y + 15y Combine like terms: -13y + 15y = 2y -5 + 2y + 5y2 = -15y + 15y Combine like terms: -15y + 15y = 0 -5 + 2y + 5y2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -1 + 0.4y + y2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 0.4y + 1 + y2 = 0 + 1 Reorder the terms: -1 + 1 + 0.4y + y2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 0.4y + y2 = 0 + 1 0.4y + y2 = 0 + 1 Combine like terms: 0 + 1 = 1 0.4y + y2 = 1 The y term is 0.4y. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4y + 0.04 + y2 = 1 + 0.04 Reorder the terms: 0.04 + 0.4y + y2 = 1 + 0.04 Combine like terms: 1 + 0.04 = 1.04 0.04 + 0.4y + y2 = 1.04 Factor a perfect square on the left side: (y + 0.2)(y + 0.2) = 1.04 Calculate the square root of the right side: 1.019803903 Break this problem into two subproblems by setting (y + 0.2) equal to 1.019803903 and -1.019803903.Subproblem 1
y + 0.2 = 1.019803903 Simplifying y + 0.2 = 1.019803903 Reorder the terms: 0.2 + y = 1.019803903 Solving 0.2 + y = 1.019803903 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + y = 1.019803903 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + y = 1.019803903 + -0.2 y = 1.019803903 + -0.2 Combine like terms: 1.019803903 + -0.2 = 0.819803903 y = 0.819803903 Simplifying y = 0.819803903Subproblem 2
y + 0.2 = -1.019803903 Simplifying y + 0.2 = -1.019803903 Reorder the terms: 0.2 + y = -1.019803903 Solving 0.2 + y = -1.019803903 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + y = -1.019803903 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + y = -1.019803903 + -0.2 y = -1.019803903 + -0.2 Combine like terms: -1.019803903 + -0.2 = -1.219803903 y = -1.219803903 Simplifying y = -1.219803903Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.819803903, -1.219803903}
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