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Simplifying 4y2 + -5y + 16 + -14y2 + -5y + 7 = 0 Reorder the terms: 16 + 7 + -5y + -5y + 4y2 + -14y2 = 0 Combine like terms: 16 + 7 = 23 23 + -5y + -5y + 4y2 + -14y2 = 0 Combine like terms: -5y + -5y = -10y 23 + -10y + 4y2 + -14y2 = 0 Combine like terms: 4y2 + -14y2 = -10y2 23 + -10y + -10y2 = 0 Solving 23 + -10y + -10y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by -10 the coefficient of the squared term: Divide each side by '-10'. -2.3 + y + y2 = 0.0 Move the constant term to the right: Add '2.3' to each side of the equation. -2.3 + y + 2.3 + y2 = 0.0 + 2.3 Reorder the terms: -2.3 + 2.3 + y + y2 = 0.0 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + y + y2 = 0.0 + 2.3 y + y2 = 0.0 + 2.3 Combine like terms: 0.0 + 2.3 = 2.3 y + y2 = 2.3 The y term is y. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + y2 = 2.3 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + y2 = 2.3 + 0.25 Combine like terms: 2.3 + 0.25 = 2.55 1.25 + y2 = 2.55 Factor a perfect square on the left side: (y + 0.5)(y + 0.5) = 2.55 Calculate the square root of the right side: 1.596871942 Break this problem into two subproblems by setting (y + 0.5) equal to 1.596871942 and -1.596871942.Subproblem 1
y + 0.5 = 1.596871942 Simplifying y + 0.5 = 1.596871942 Reorder the terms: 0.5 + y = 1.596871942 Solving 0.5 + y = 1.596871942 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = 1.596871942 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = 1.596871942 + -0.5 y = 1.596871942 + -0.5 Combine like terms: 1.596871942 + -0.5 = 1.096871942 y = 1.096871942 Simplifying y = 1.096871942Subproblem 2
y + 0.5 = -1.596871942 Simplifying y + 0.5 = -1.596871942 Reorder the terms: 0.5 + y = -1.596871942 Solving 0.5 + y = -1.596871942 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + y = -1.596871942 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + y = -1.596871942 + -0.5 y = -1.596871942 + -0.5 Combine like terms: -1.596871942 + -0.5 = -2.096871942 y = -2.096871942 Simplifying y = -2.096871942Solution
The solution to the problem is based on the solutions from the subproblems. y = {1.096871942, -2.096871942}
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