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Simplifying 17y + -13 + 6y2 + -14y + 5 = 0 Reorder the terms: -13 + 5 + 17y + -14y + 6y2 = 0 Combine like terms: -13 + 5 = -8 -8 + 17y + -14y + 6y2 = 0 Combine like terms: 17y + -14y = 3y -8 + 3y + 6y2 = 0 Solving -8 + 3y + 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 + 0.5y + y2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 0.5y + 1.333333333 + y2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 0.5y + y2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 0.5y + y2 = 0 + 1.333333333 0.5y + y2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 0.5y + y2 = 1.333333333 The y term is 0.5y. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5y + 0.0625 + y2 = 1.333333333 + 0.0625 Reorder the terms: 0.0625 + 0.5y + y2 = 1.333333333 + 0.0625 Combine like terms: 1.333333333 + 0.0625 = 1.395833333 0.0625 + 0.5y + y2 = 1.395833333 Factor a perfect square on the left side: (y + 0.25)(y + 0.25) = 1.395833333 Calculate the square root of the right side: 1.181453906 Break this problem into two subproblems by setting (y + 0.25) equal to 1.181453906 and -1.181453906.Subproblem 1
y + 0.25 = 1.181453906 Simplifying y + 0.25 = 1.181453906 Reorder the terms: 0.25 + y = 1.181453906 Solving 0.25 + y = 1.181453906 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = 1.181453906 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = 1.181453906 + -0.25 y = 1.181453906 + -0.25 Combine like terms: 1.181453906 + -0.25 = 0.931453906 y = 0.931453906 Simplifying y = 0.931453906Subproblem 2
y + 0.25 = -1.181453906 Simplifying y + 0.25 = -1.181453906 Reorder the terms: 0.25 + y = -1.181453906 Solving 0.25 + y = -1.181453906 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + y = -1.181453906 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + y = -1.181453906 + -0.25 y = -1.181453906 + -0.25 Combine like terms: -1.181453906 + -0.25 = -1.431453906 y = -1.431453906 Simplifying y = -1.431453906Solution
The solution to the problem is based on the solutions from the subproblems. y = {0.931453906, -1.431453906}
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