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Simplifying 2y2 + 3y + -24 = 0 Reorder the terms: -24 + 3y + 2y2 = 0 Solving -24 + 3y + 2y2 = 0 Solving for variable 'y'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -12 + 1.5y + y2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 1.5y + 12 + y2 = 0 + 12 Reorder the terms: -12 + 12 + 1.5y + y2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 1.5y + y2 = 0 + 12 1.5y + y2 = 0 + 12 Combine like terms: 0 + 12 = 12 1.5y + y2 = 12 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 = 12 + 0.5625 Reorder the terms: 0.5625 + 1.5y + y2 = 12 + 0.5625 Combine like terms: 12 + 0.5625 = 12.5625 0.5625 + 1.5y + y2 = 12.5625 Factor a perfect square on the left side: (y + 0.75)(y + 0.75) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (y + 0.75) equal to 3.54436172 and -3.54436172.Subproblem 1
y + 0.75 = 3.54436172 Simplifying y + 0.75 = 3.54436172 Reorder the terms: 0.75 + y = 3.54436172 Solving 0.75 + y = 3.54436172 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 = 3.54436172 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = 3.54436172 + -0.75 y = 3.54436172 + -0.75 Combine like terms: 3.54436172 + -0.75 = 2.79436172 y = 2.79436172 Simplifying y = 2.79436172Subproblem 2
y + 0.75 = -3.54436172 Simplifying y + 0.75 = -3.54436172 Reorder the terms: 0.75 + y = -3.54436172 Solving 0.75 + y = -3.54436172 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 = -3.54436172 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + y = -3.54436172 + -0.75 y = -3.54436172 + -0.75 Combine like terms: -3.54436172 + -0.75 = -4.29436172 y = -4.29436172 Simplifying y = -4.29436172Solution
The solution to the problem is based on the solutions from the subproblems. y = {2.79436172, -4.29436172}
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