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Simplifying 4y2 + -4y + -23 = 0 Reorder the terms: -23 + -4y + 4y2 = 0 Solving -23 + -4y + 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 + -1y + y2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + -1y + 5.75 + y2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + -1y + y2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + -1y + y2 = 0 + 5.75 -1y + y2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 -1y + y2 = 5.75 The y term is -1y. 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. -1y + 0.25 + y2 = 5.75 + 0.25 Reorder the terms: 0.25 + -1y + y2 = 5.75 + 0.25 Combine like terms: 5.75 + 0.25 = 6 0.25 + -1y + y2 = 6 Factor a perfect square on the left side: (y + -0.5)(y + -0.5) = 6 Calculate the square root of the right side: 2.449489743 Break this problem into two subproblems by setting (y + -0.5) equal to 2.449489743 and -2.449489743.Subproblem 1
y + -0.5 = 2.449489743 Simplifying y + -0.5 = 2.449489743 Reorder the terms: -0.5 + y = 2.449489743 Solving -0.5 + y = 2.449489743 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 = 2.449489743 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = 2.449489743 + 0.5 y = 2.449489743 + 0.5 Combine like terms: 2.449489743 + 0.5 = 2.949489743 y = 2.949489743 Simplifying y = 2.949489743Subproblem 2
y + -0.5 = -2.449489743 Simplifying y + -0.5 = -2.449489743 Reorder the terms: -0.5 + y = -2.449489743 Solving -0.5 + y = -2.449489743 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 = -2.449489743 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + y = -2.449489743 + 0.5 y = -2.449489743 + 0.5 Combine like terms: -2.449489743 + 0.5 = -1.949489743 y = -1.949489743 Simplifying y = -1.949489743Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.949489743, -1.949489743}
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