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Simplifying 2x2 + 3x + 4 = 84 Reorder the terms: 4 + 3x + 2x2 = 84 Solving 4 + 3x + 2x2 = 84 Solving for variable 'x'. Reorder the terms: 4 + -84 + 3x + 2x2 = 84 + -84 Combine like terms: 4 + -84 = -80 -80 + 3x + 2x2 = 84 + -84 Combine like terms: 84 + -84 = 0 -80 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -40 + 1.5x + x2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + 1.5x + 40 + x2 = 0 + 40 Reorder the terms: -40 + 40 + 1.5x + x2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 1.5x + x2 = 0 + 40 1.5x + x2 = 0 + 40 Combine like terms: 0 + 40 = 40 1.5x + x2 = 40 The x term is 1.5x. 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. 1.5x + 0.5625 + x2 = 40 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 40 + 0.5625 Combine like terms: 40 + 0.5625 = 40.5625 0.5625 + 1.5x + x2 = 40.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 40.5625 Calculate the square root of the right side: 6.368869601 Break this problem into two subproblems by setting (x + 0.75) equal to 6.368869601 and -6.368869601.Subproblem 1
x + 0.75 = 6.368869601 Simplifying x + 0.75 = 6.368869601 Reorder the terms: 0.75 + x = 6.368869601 Solving 0.75 + x = 6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 6.368869601 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 6.368869601 + -0.75 x = 6.368869601 + -0.75 Combine like terms: 6.368869601 + -0.75 = 5.618869601 x = 5.618869601 Simplifying x = 5.618869601Subproblem 2
x + 0.75 = -6.368869601 Simplifying x + 0.75 = -6.368869601 Reorder the terms: 0.75 + x = -6.368869601 Solving 0.75 + x = -6.368869601 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -6.368869601 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -6.368869601 + -0.75 x = -6.368869601 + -0.75 Combine like terms: -6.368869601 + -0.75 = -7.118869601 x = -7.118869601 Simplifying x = -7.118869601Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.618869601, -7.118869601}
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