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Simplifying 4y2 + -20y + 23 = 0 Reorder the terms: 23 + -20y + 4y2 = 0 Solving 23 + -20y + 4y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.75 + -5y + y2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + -5y + -5.75 + y2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + -5y + y2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + -5y + y2 = 0 + -5.75 -5y + y2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 -5y + y2 = -5.75 The y term is -5y. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5y + 6.25 + y2 = -5.75 + 6.25 Reorder the terms: 6.25 + -5y + y2 = -5.75 + 6.25 Combine like terms: -5.75 + 6.25 = 0.5 6.25 + -5y + y2 = 0.5 Factor a perfect square on the left side: (y + -2.5)(y + -2.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (y + -2.5) equal to 0.707106781 and -0.707106781.Subproblem 1
y + -2.5 = 0.707106781 Simplifying y + -2.5 = 0.707106781 Reorder the terms: -2.5 + y = 0.707106781 Solving -2.5 + y = 0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = 0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = 0.707106781 + 2.5 y = 0.707106781 + 2.5 Combine like terms: 0.707106781 + 2.5 = 3.207106781 y = 3.207106781 Simplifying y = 3.207106781Subproblem 2
y + -2.5 = -0.707106781 Simplifying y + -2.5 = -0.707106781 Reorder the terms: -2.5 + y = -0.707106781 Solving -2.5 + y = -0.707106781 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + y = -0.707106781 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + y = -0.707106781 + 2.5 y = -0.707106781 + 2.5 Combine like terms: -0.707106781 + 2.5 = 1.792893219 y = 1.792893219 Simplifying y = 1.792893219Solution
The solution to the problem is based on the solutions from the subproblems. y = {3.207106781, 1.792893219}
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