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Simplifying 12.23 + 0.06x(x + 3) = 12.98 + -0.12x Reorder the terms: 12.23 + 0.06x(3 + x) = 12.98 + -0.12x 12.23 + (3 * 0.06x + x * 0.06x) = 12.98 + -0.12x 12.23 + (0.18x + 0.06x2) = 12.98 + -0.12x Solving 12.23 + 0.18x + 0.06x2 = 12.98 + -0.12x Solving for variable 'x'. Reorder the terms: 12.23 + -12.98 + 0.18x + 0.12x + 0.06x2 = 12.98 + -0.12x + -12.98 + 0.12x Combine like terms: 12.23 + -12.98 = -0.75 -0.75 + 0.18x + 0.12x + 0.06x2 = 12.98 + -0.12x + -12.98 + 0.12x Combine like terms: 0.18x + 0.12x = 0.3x -0.75 + 0.3x + 0.06x2 = 12.98 + -0.12x + -12.98 + 0.12x Reorder the terms: -0.75 + 0.3x + 0.06x2 = 12.98 + -12.98 + -0.12x + 0.12x Combine like terms: 12.98 + -12.98 = 0.00 -0.75 + 0.3x + 0.06x2 = 0.00 + -0.12x + 0.12x -0.75 + 0.3x + 0.06x2 = -0.12x + 0.12x Combine like terms: -0.12x + 0.12x = 0.00 -0.75 + 0.3x + 0.06x2 = 0.00 Begin completing the square. Divide all terms by 0.06 the coefficient of the squared term: Divide each side by '0.06'. -12.5 + 5x + x2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 5x + 12.5 + x2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 5x + x2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 5x + x2 = 0 + 12.5 5x + x2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 5x + x2 = 12.5 The x term is 5x. 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. 5x + 6.25 + x2 = 12.5 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 12.5 + 6.25 Combine like terms: 12.5 + 6.25 = 18.75 6.25 + 5x + x2 = 18.75 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 18.75 Calculate the square root of the right side: 4.330127019 Break this problem into two subproblems by setting (x + 2.5) equal to 4.330127019 and -4.330127019.Subproblem 1
x + 2.5 = 4.330127019 Simplifying x + 2.5 = 4.330127019 Reorder the terms: 2.5 + x = 4.330127019 Solving 2.5 + x = 4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 4.330127019 + -2.5 x = 4.330127019 + -2.5 Combine like terms: 4.330127019 + -2.5 = 1.830127019 x = 1.830127019 Simplifying x = 1.830127019Subproblem 2
x + 2.5 = -4.330127019 Simplifying x + 2.5 = -4.330127019 Reorder the terms: 2.5 + x = -4.330127019 Solving 2.5 + x = -4.330127019 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -4.330127019 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -4.330127019 + -2.5 x = -4.330127019 + -2.5 Combine like terms: -4.330127019 + -2.5 = -6.830127019 x = -6.830127019 Simplifying x = -6.830127019Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.830127019, -6.830127019}
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