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Simplifying (6x2 + -3x + 11) + (2x2 + 6x + -67) = 0 Reorder the terms: (11 + -3x + 6x2) + (2x2 + 6x + -67) = 0 Remove parenthesis around (11 + -3x + 6x2) 11 + -3x + 6x2 + (2x2 + 6x + -67) = 0 Reorder the terms: 11 + -3x + 6x2 + (-67 + 6x + 2x2) = 0 Remove parenthesis around (-67 + 6x + 2x2) 11 + -3x + 6x2 + -67 + 6x + 2x2 = 0 Reorder the terms: 11 + -67 + -3x + 6x + 6x2 + 2x2 = 0 Combine like terms: 11 + -67 = -56 -56 + -3x + 6x + 6x2 + 2x2 = 0 Combine like terms: -3x + 6x = 3x -56 + 3x + 6x2 + 2x2 = 0 Combine like terms: 6x2 + 2x2 = 8x2 -56 + 3x + 8x2 = 0 Solving -56 + 3x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -7 + 0.375x + x2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 0.375x + 7 + x2 = 0 + 7 Reorder the terms: -7 + 7 + 0.375x + x2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 0.375x + x2 = 0 + 7 0.375x + x2 = 0 + 7 Combine like terms: 0 + 7 = 7 0.375x + x2 = 7 The x term is 0.375x. Take half its coefficient (0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. 0.375x + 0.03515625 + x2 = 7 + 0.03515625 Reorder the terms: 0.03515625 + 0.375x + x2 = 7 + 0.03515625 Combine like terms: 7 + 0.03515625 = 7.03515625 0.03515625 + 0.375x + x2 = 7.03515625 Factor a perfect square on the left side: (x + 0.1875)(x + 0.1875) = 7.03515625 Calculate the square root of the right side: 2.652386897 Break this problem into two subproblems by setting (x + 0.1875) equal to 2.652386897 and -2.652386897.Subproblem 1
x + 0.1875 = 2.652386897 Simplifying x + 0.1875 = 2.652386897 Reorder the terms: 0.1875 + x = 2.652386897 Solving 0.1875 + x = 2.652386897 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = 2.652386897 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = 2.652386897 + -0.1875 x = 2.652386897 + -0.1875 Combine like terms: 2.652386897 + -0.1875 = 2.464886897 x = 2.464886897 Simplifying x = 2.464886897Subproblem 2
x + 0.1875 = -2.652386897 Simplifying x + 0.1875 = -2.652386897 Reorder the terms: 0.1875 + x = -2.652386897 Solving 0.1875 + x = -2.652386897 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1875' to each side of the equation. 0.1875 + -0.1875 + x = -2.652386897 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + x = -2.652386897 + -0.1875 x = -2.652386897 + -0.1875 Combine like terms: -2.652386897 + -0.1875 = -2.839886897 x = -2.839886897 Simplifying x = -2.839886897Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.464886897, -2.839886897}
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