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Simplifying (7x4 + 9x3 + -6x2 + 10) + -1(7x4 + 9x3 + -4x2 + 3x + -6) = 0 Reorder the terms: (10 + -6x2 + 9x3 + 7x4) + -1(7x4 + 9x3 + -4x2 + 3x + -6) = 0 Remove parenthesis around (10 + -6x2 + 9x3 + 7x4) 10 + -6x2 + 9x3 + 7x4 + -1(7x4 + 9x3 + -4x2 + 3x + -6) = 0 Reorder the terms: 10 + -6x2 + 9x3 + 7x4 + -1(-6 + 3x + -4x2 + 9x3 + 7x4) = 0 10 + -6x2 + 9x3 + 7x4 + (-6 * -1 + 3x * -1 + -4x2 * -1 + 9x3 * -1 + 7x4 * -1) = 0 10 + -6x2 + 9x3 + 7x4 + (6 + -3x + 4x2 + -9x3 + -7x4) = 0 Reorder the terms: 10 + 6 + -3x + -6x2 + 4x2 + 9x3 + -9x3 + 7x4 + -7x4 = 0 Combine like terms: 10 + 6 = 16 16 + -3x + -6x2 + 4x2 + 9x3 + -9x3 + 7x4 + -7x4 = 0 Combine like terms: -6x2 + 4x2 = -2x2 16 + -3x + -2x2 + 9x3 + -9x3 + 7x4 + -7x4 = 0 Combine like terms: 9x3 + -9x3 = 0 16 + -3x + -2x2 + 0 + 7x4 + -7x4 = 0 16 + -3x + -2x2 + 7x4 + -7x4 = 0 Combine like terms: 7x4 + -7x4 = 0 16 + -3x + -2x2 + 0 = 0 16 + -3x + -2x2 = 0 Solving 16 + -3x + -2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -8 + 1.5x + x2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 1.5x + 8 + x2 = 0 + 8 Reorder the terms: -8 + 8 + 1.5x + x2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 1.5x + x2 = 0 + 8 1.5x + x2 = 0 + 8 Combine like terms: 0 + 8 = 8 1.5x + x2 = 8 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 = 8 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 8 + 0.5625 Combine like terms: 8 + 0.5625 = 8.5625 0.5625 + 1.5x + x2 = 8.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 8.5625 Calculate the square root of the right side: 2.926174978 Break this problem into two subproblems by setting (x + 0.75) equal to 2.926174978 and -2.926174978.Subproblem 1
x + 0.75 = 2.926174978 Simplifying x + 0.75 = 2.926174978 Reorder the terms: 0.75 + x = 2.926174978 Solving 0.75 + x = 2.926174978 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 = 2.926174978 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 2.926174978 + -0.75 x = 2.926174978 + -0.75 Combine like terms: 2.926174978 + -0.75 = 2.176174978 x = 2.176174978 Simplifying x = 2.176174978Subproblem 2
x + 0.75 = -2.926174978 Simplifying x + 0.75 = -2.926174978 Reorder the terms: 0.75 + x = -2.926174978 Solving 0.75 + x = -2.926174978 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 = -2.926174978 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -2.926174978 + -0.75 x = -2.926174978 + -0.75 Combine like terms: -2.926174978 + -0.75 = -3.676174978 x = -3.676174978 Simplifying x = -3.676174978Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.176174978, -3.676174978}
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