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Simplifying (5x2 + -3x + 22) + (6x2 + 6x + -67) = 0 Reorder the terms: (22 + -3x + 5x2) + (6x2 + 6x + -67) = 0 Remove parenthesis around (22 + -3x + 5x2) 22 + -3x + 5x2 + (6x2 + 6x + -67) = 0 Reorder the terms: 22 + -3x + 5x2 + (-67 + 6x + 6x2) = 0 Remove parenthesis around (-67 + 6x + 6x2) 22 + -3x + 5x2 + -67 + 6x + 6x2 = 0 Reorder the terms: 22 + -67 + -3x + 6x + 5x2 + 6x2 = 0 Combine like terms: 22 + -67 = -45 -45 + -3x + 6x + 5x2 + 6x2 = 0 Combine like terms: -3x + 6x = 3x -45 + 3x + 5x2 + 6x2 = 0 Combine like terms: 5x2 + 6x2 = 11x2 -45 + 3x + 11x2 = 0 Solving -45 + 3x + 11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -4.090909091 + 0.2727272727x + x2 = 0 Move the constant term to the right: Add '4.090909091' to each side of the equation. -4.090909091 + 0.2727272727x + 4.090909091 + x2 = 0 + 4.090909091 Reorder the terms: -4.090909091 + 4.090909091 + 0.2727272727x + x2 = 0 + 4.090909091 Combine like terms: -4.090909091 + 4.090909091 = 0.000000000 0.000000000 + 0.2727272727x + x2 = 0 + 4.090909091 0.2727272727x + x2 = 0 + 4.090909091 Combine like terms: 0 + 4.090909091 = 4.090909091 0.2727272727x + x2 = 4.090909091 The x term is 0.2727272727x. Take half its coefficient (0.1363636364). Square it (0.01859504133) and add it to both sides. Add '0.01859504133' to each side of the equation. 0.2727272727x + 0.01859504133 + x2 = 4.090909091 + 0.01859504133 Reorder the terms: 0.01859504133 + 0.2727272727x + x2 = 4.090909091 + 0.01859504133 Combine like terms: 4.090909091 + 0.01859504133 = 4.10950413233 0.01859504133 + 0.2727272727x + x2 = 4.10950413233 Factor a perfect square on the left side: (x + 0.1363636364)(x + 0.1363636364) = 4.10950413233 Calculate the square root of the right side: 2.027191193 Break this problem into two subproblems by setting (x + 0.1363636364) equal to 2.027191193 and -2.027191193.Subproblem 1
x + 0.1363636364 = 2.027191193 Simplifying x + 0.1363636364 = 2.027191193 Reorder the terms: 0.1363636364 + x = 2.027191193 Solving 0.1363636364 + x = 2.027191193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = 2.027191193 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = 2.027191193 + -0.1363636364 x = 2.027191193 + -0.1363636364 Combine like terms: 2.027191193 + -0.1363636364 = 1.8908275566 x = 1.8908275566 Simplifying x = 1.8908275566Subproblem 2
x + 0.1363636364 = -2.027191193 Simplifying x + 0.1363636364 = -2.027191193 Reorder the terms: 0.1363636364 + x = -2.027191193 Solving 0.1363636364 + x = -2.027191193 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1363636364' to each side of the equation. 0.1363636364 + -0.1363636364 + x = -2.027191193 + -0.1363636364 Combine like terms: 0.1363636364 + -0.1363636364 = 0.0000000000 0.0000000000 + x = -2.027191193 + -0.1363636364 x = -2.027191193 + -0.1363636364 Combine like terms: -2.027191193 + -0.1363636364 = -2.1635548294 x = -2.1635548294 Simplifying x = -2.1635548294Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.8908275566, -2.1635548294}
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