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Simplifying x2 + (-0.75x + 1.25)(-0.75x + 1.25) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 Reorder the terms: x2 + (1.25 + -0.75x)(-0.75x + 1.25) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 Reorder the terms: x2 + (1.25 + -0.75x)(1.25 + -0.75x) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 Multiply (1.25 + -0.75x) * (1.25 + -0.75x) x2 + (1.25(1.25 + -0.75x) + -0.75x * (1.25 + -0.75x)) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + ((1.25 * 1.25 + -0.75x * 1.25) + -0.75x * (1.25 + -0.75x)) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + ((1.5625 + -0.9375x) + -0.75x * (1.25 + -0.75x)) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + (1.5625 + -0.9375x + (1.25 * -0.75x + -0.75x * -0.75x)) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + (1.5625 + -0.9375x + (-0.9375x + 0.5625x2)) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 Combine like terms: -0.9375x + -0.9375x = -1.875x x2 + (1.5625 + -1.875x + 0.5625x2) + 4x(-0.75x + 1.25) + -12x + -6(-0.75x + 1.25) + 5 = 0 Reorder the terms: x2 + 1.5625 + -1.875x + 0.5625x2 + 4x(1.25 + -0.75x) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + 1.5625 + -1.875x + 0.5625x2 + (1.25 * 4x + -0.75x * 4x) + -12x + -6(-0.75x + 1.25) + 5 = 0 x2 + 1.5625 + -1.875x + 0.5625x2 + (5x + -3x2) + -12x + -6(-0.75x + 1.25) + 5 = 0 Reorder the terms: x2 + 1.5625 + -1.875x + 0.5625x2 + 5x + -3x2 + -12x + -6(1.25 + -0.75x) + 5 = 0 x2 + 1.5625 + -1.875x + 0.5625x2 + 5x + -3x2 + -12x + (1.25 * -6 + -0.75x * -6) + 5 = 0 x2 + 1.5625 + -1.875x + 0.5625x2 + 5x + -3x2 + -12x + (-7.5 + 4.5x) + 5 = 0 Reorder the terms: 1.5625 + -7.5 + 5 + -1.875x + 5x + -12x + 4.5x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: 1.5625 + -7.5 = -5.9375 -5.9375 + 5 + -1.875x + 5x + -12x + 4.5x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: -5.9375 + 5 = -0.9375 -0.9375 + -1.875x + 5x + -12x + 4.5x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: -1.875x + 5x = 3.125x -0.9375 + 3.125x + -12x + 4.5x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: 3.125x + -12x = -8.875x -0.9375 + -8.875x + 4.5x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: -8.875x + 4.5x = -4.375x -0.9375 + -4.375x + x2 + 0.5625x2 + -3x2 = 0 Combine like terms: x2 + 0.5625x2 = 1.5625x2 -0.9375 + -4.375x + 1.5625x2 + -3x2 = 0 Combine like terms: 1.5625x2 + -3x2 = -1.4375x2 -0.9375 + -4.375x + -1.4375x2 = 0 Solving -0.9375 + -4.375x + -1.4375x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -1.4375 the coefficient of the squared term: Divide each side by '-1.4375'. 0.652173913 + 3.043478261x + x2 = 0 Move the constant term to the right: Add '-0.652173913' to each side of the equation. 0.652173913 + 3.043478261x + -0.652173913 + x2 = 0 + -0.652173913 Reorder the terms: 0.652173913 + -0.652173913 + 3.043478261x + x2 = 0 + -0.652173913 Combine like terms: 0.652173913 + -0.652173913 = 0.000000000 0.000000000 + 3.043478261x + x2 = 0 + -0.652173913 3.043478261x + x2 = 0 + -0.652173913 Combine like terms: 0 + -0.652173913 = -0.652173913 3.043478261x + x2 = -0.652173913 The x term is 3.043478261x. Take half its coefficient (1.521739131). Square it (2.315689983) and add it to both sides. Add '2.315689983' to each side of the equation. 3.043478261x + 2.315689983 + x2 = -0.652173913 + 2.315689983 Reorder the terms: 2.315689983 + 3.043478261x + x2 = -0.652173913 + 2.315689983 Combine like terms: -0.652173913 + 2.315689983 = 1.66351607 2.315689983 + 3.043478261x + x2 = 1.66351607 Factor a perfect square on the left side: (x + 1.521739131)(x + 1.521739131) = 1.66351607 Calculate the square root of the right side: 1.289773651 Break this problem into two subproblems by setting (x + 1.521739131) equal to 1.289773651 and -1.289773651.Subproblem 1
x + 1.521739131 = 1.289773651 Simplifying x + 1.521739131 = 1.289773651 Reorder the terms: 1.521739131 + x = 1.289773651 Solving 1.521739131 + x = 1.289773651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.521739131' to each side of the equation. 1.521739131 + -1.521739131 + x = 1.289773651 + -1.521739131 Combine like terms: 1.521739131 + -1.521739131 = 0.000000000 0.000000000 + x = 1.289773651 + -1.521739131 x = 1.289773651 + -1.521739131 Combine like terms: 1.289773651 + -1.521739131 = -0.23196548 x = -0.23196548 Simplifying x = -0.23196548Subproblem 2
x + 1.521739131 = -1.289773651 Simplifying x + 1.521739131 = -1.289773651 Reorder the terms: 1.521739131 + x = -1.289773651 Solving 1.521739131 + x = -1.289773651 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.521739131' to each side of the equation. 1.521739131 + -1.521739131 + x = -1.289773651 + -1.521739131 Combine like terms: 1.521739131 + -1.521739131 = 0.000000000 0.000000000 + x = -1.289773651 + -1.521739131 x = -1.289773651 + -1.521739131 Combine like terms: -1.289773651 + -1.521739131 = -2.811512782 x = -2.811512782 Simplifying x = -2.811512782Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.23196548, -2.811512782}
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