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Simplifying x(4x + 3) + -1 = 24 + x Reorder the terms: x(3 + 4x) + -1 = 24 + x (3 * x + 4x * x) + -1 = 24 + x (3x + 4x2) + -1 = 24 + x Reorder the terms: -1 + 3x + 4x2 = 24 + x Solving -1 + 3x + 4x2 = 24 + x Solving for variable 'x'. Reorder the terms: -1 + -24 + 3x + -1x + 4x2 = 24 + x + -24 + -1x Combine like terms: -1 + -24 = -25 -25 + 3x + -1x + 4x2 = 24 + x + -24 + -1x Combine like terms: 3x + -1x = 2x -25 + 2x + 4x2 = 24 + x + -24 + -1x Reorder the terms: -25 + 2x + 4x2 = 24 + -24 + x + -1x Combine like terms: 24 + -24 = 0 -25 + 2x + 4x2 = 0 + x + -1x -25 + 2x + 4x2 = x + -1x Combine like terms: x + -1x = 0 -25 + 2x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.25 + 0.5x + x2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + 0.5x + 6.25 + x2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + 0.5x + x2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + 0.5x + x2 = 0 + 6.25 0.5x + x2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 0.5x + x2 = 6.25 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 6.25 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 6.25 + 0.0625 Combine like terms: 6.25 + 0.0625 = 6.3125 0.0625 + 0.5x + x2 = 6.3125 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 6.3125 Calculate the square root of the right side: 2.512468905 Break this problem into two subproblems by setting (x + 0.25) equal to 2.512468905 and -2.512468905.Subproblem 1
x + 0.25 = 2.512468905 Simplifying x + 0.25 = 2.512468905 Reorder the terms: 0.25 + x = 2.512468905 Solving 0.25 + x = 2.512468905 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 2.512468905 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 2.512468905 + -0.25 x = 2.512468905 + -0.25 Combine like terms: 2.512468905 + -0.25 = 2.262468905 x = 2.262468905 Simplifying x = 2.262468905Subproblem 2
x + 0.25 = -2.512468905 Simplifying x + 0.25 = -2.512468905 Reorder the terms: 0.25 + x = -2.512468905 Solving 0.25 + x = -2.512468905 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -2.512468905 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -2.512468905 + -0.25 x = -2.512468905 + -0.25 Combine like terms: -2.512468905 + -0.25 = -2.762468905 x = -2.762468905 Simplifying x = -2.762468905Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.262468905, -2.762468905}
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