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Simplifying 13 + -1(2x + 2) = 2(2 + x) * 3x Reorder the terms: 13 + -1(2 + 2x) = 2(2 + x) * 3x 13 + (2 * -1 + 2x * -1) = 2(2 + x) * 3x 13 + (-2 + -2x) = 2(2 + x) * 3x Combine like terms: 13 + -2 = 11 11 + -2x = 2(2 + x) * 3x Reorder the terms for easier multiplication: 11 + -2x = 2 * 3x(2 + x) Multiply 2 * 3 11 + -2x = 6x(2 + x) 11 + -2x = (2 * 6x + x * 6x) 11 + -2x = (12x + 6x2) Solving 11 + -2x = 12x + 6x2 Solving for variable 'x'. Combine like terms: -2x + -12x = -14x 11 + -14x + -6x2 = 12x + 6x2 + -12x + -6x2 Reorder the terms: 11 + -14x + -6x2 = 12x + -12x + 6x2 + -6x2 Combine like terms: 12x + -12x = 0 11 + -14x + -6x2 = 0 + 6x2 + -6x2 11 + -14x + -6x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 11 + -14x + -6x2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1.833333333 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '1.833333333' to each side of the equation. -1.833333333 + 2.333333333x + 1.833333333 + x2 = 0 + 1.833333333 Reorder the terms: -1.833333333 + 1.833333333 + 2.333333333x + x2 = 0 + 1.833333333 Combine like terms: -1.833333333 + 1.833333333 = 0.000000000 0.000000000 + 2.333333333x + x2 = 0 + 1.833333333 2.333333333x + x2 = 0 + 1.833333333 Combine like terms: 0 + 1.833333333 = 1.833333333 2.333333333x + x2 = 1.833333333 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 1.833333333 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 1.833333333 + 1.361111112 Combine like terms: 1.833333333 + 1.361111112 = 3.194444445 1.361111112 + 2.333333333x + x2 = 3.194444445 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 3.194444445 Calculate the square root of the right side: 1.787300883 Break this problem into two subproblems by setting (x + 1.166666667) equal to 1.787300883 and -1.787300883.Subproblem 1
x + 1.166666667 = 1.787300883 Simplifying x + 1.166666667 = 1.787300883 Reorder the terms: 1.166666667 + x = 1.787300883 Solving 1.166666667 + x = 1.787300883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 1.787300883 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 1.787300883 + -1.166666667 x = 1.787300883 + -1.166666667 Combine like terms: 1.787300883 + -1.166666667 = 0.620634216 x = 0.620634216 Simplifying x = 0.620634216Subproblem 2
x + 1.166666667 = -1.787300883 Simplifying x + 1.166666667 = -1.787300883 Reorder the terms: 1.166666667 + x = -1.787300883 Solving 1.166666667 + x = -1.787300883 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -1.787300883 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -1.787300883 + -1.166666667 x = -1.787300883 + -1.166666667 Combine like terms: -1.787300883 + -1.166666667 = -2.95396755 x = -2.95396755 Simplifying x = -2.95396755Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.620634216, -2.95396755}
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