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Simplifying (6 + 4x2 + -5x) + (7x + 5x2 + -12) = 0 Reorder the terms: (6 + -5x + 4x2) + (7x + 5x2 + -12) = 0 Remove parenthesis around (6 + -5x + 4x2) 6 + -5x + 4x2 + (7x + 5x2 + -12) = 0 Reorder the terms: 6 + -5x + 4x2 + (-12 + 7x + 5x2) = 0 Remove parenthesis around (-12 + 7x + 5x2) 6 + -5x + 4x2 + -12 + 7x + 5x2 = 0 Reorder the terms: 6 + -12 + -5x + 7x + 4x2 + 5x2 = 0 Combine like terms: 6 + -12 = -6 -6 + -5x + 7x + 4x2 + 5x2 = 0 Combine like terms: -5x + 7x = 2x -6 + 2x + 4x2 + 5x2 = 0 Combine like terms: 4x2 + 5x2 = 9x2 -6 + 2x + 9x2 = 0 Solving -6 + 2x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.6666666667 + 0.2222222222x + x2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + 0.2222222222x + 0.6666666667 + x2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + 0.2222222222x + x2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + 0.2222222222x + x2 = 0 + 0.6666666667 0.2222222222x + x2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 0.2222222222x + x2 = 0.6666666667 The x term is 0.2222222222x. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222x + 0.01234567901 + x2 = 0.6666666667 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222x + x2 = 0.6666666667 + 0.01234567901 Combine like terms: 0.6666666667 + 0.01234567901 = 0.67901234571 0.01234567901 + 0.2222222222x + x2 = 0.67901234571 Factor a perfect square on the left side: (x + 0.1111111111)(x + 0.1111111111) = 0.67901234571 Calculate the square root of the right side: 0.824022054 Break this problem into two subproblems by setting (x + 0.1111111111) equal to 0.824022054 and -0.824022054.Subproblem 1
x + 0.1111111111 = 0.824022054 Simplifying x + 0.1111111111 = 0.824022054 Reorder the terms: 0.1111111111 + x = 0.824022054 Solving 0.1111111111 + x = 0.824022054 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + x = 0.824022054 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = 0.824022054 + -0.1111111111 x = 0.824022054 + -0.1111111111 Combine like terms: 0.824022054 + -0.1111111111 = 0.7129109429 x = 0.7129109429 Simplifying x = 0.7129109429Subproblem 2
x + 0.1111111111 = -0.824022054 Simplifying x + 0.1111111111 = -0.824022054 Reorder the terms: 0.1111111111 + x = -0.824022054 Solving 0.1111111111 + x = -0.824022054 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + x = -0.824022054 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + x = -0.824022054 + -0.1111111111 x = -0.824022054 + -0.1111111111 Combine like terms: -0.824022054 + -0.1111111111 = -0.9351331651 x = -0.9351331651 Simplifying x = -0.9351331651Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7129109429, -0.9351331651}
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