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Simplifying 8y2 + 12y + 3 = 0 Reorder the terms: 3 + 12y + 8y2 = 0 Solving 3 + 12y + 8y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 0.375 + 1.5y + y2 = 0 Move the constant term to the right: Add '-0.375' to each side of the equation. 0.375 + 1.5y + -0.375 + y2 = 0 + -0.375 Reorder the terms: 0.375 + -0.375 + 1.5y + y2 = 0 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + 1.5y + y2 = 0 + -0.375 1.5y + y2 = 0 + -0.375 Combine like terms: 0 + -0.375 = -0.375 1.5y + y2 = -0.375 The y term is 1.5y. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5y + 0.5625 + y2 = -0.375 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = -0.375 + 0.5625 Combine like terms: -0.375 + 0.5625 = 0.1875 0.5625 + 1.5y + y2 = 0.1875 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 0.1875 Calculate the square root of the right side: 0.433012702 Break this problem into two subproblems by setting (y + 0.75) equal to 0.433012702 and -0.433012702.Subproblem 1
y + 0.75 = 0.433012702 Simplifying y + 0.75 = 0.433012702 Reorder the terms: 0.75 + y = 0.433012702 Solving 0.75 + y = 0.433012702 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = 0.433012702 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 0.433012702 + -0.75 y = 0.433012702 + -0.75 Combine like terms: 0.433012702 + -0.75 = -0.316987298 y = -0.316987298 Simplifying y = -0.316987298Subproblem 2
y + 0.75 = -0.433012702 Simplifying y + 0.75 = -0.433012702 Reorder the terms: 0.75 + y = -0.433012702 Solving 0.75 + y = -0.433012702 Solving for variable 'y'. Move all terms containing y to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + y = -0.433012702 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -0.433012702 + -0.75 y = -0.433012702 + -0.75 Combine like terms: -0.433012702 + -0.75 = -1.183012702 y = -1.183012702 Simplifying y = -1.183012702Solution
The solution to the problem is based on the solutions from the subproblems. y = {-0.316987298, -1.183012702}
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