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Simplifying 52x(14 + -13x) + -6x = 43 + 4x (14 * 52x + -13x * 52x) + -6x = 43 + 4x (728x + -676x2) + -6x = 43 + 4x Reorder the terms: 728x + -6x + -676x2 = 43 + 4x Combine like terms: 728x + -6x = 722x 722x + -676x2 = 43 + 4x Solving 722x + -676x2 = 43 + 4x Solving for variable 'x'. Reorder the terms: -43 + 722x + -4x + -676x2 = 43 + 4x + -43 + -4x Combine like terms: 722x + -4x = 718x -43 + 718x + -676x2 = 43 + 4x + -43 + -4x Reorder the terms: -43 + 718x + -676x2 = 43 + -43 + 4x + -4x Combine like terms: 43 + -43 = 0 -43 + 718x + -676x2 = 0 + 4x + -4x -43 + 718x + -676x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -43 + 718x + -676x2 = 0 Begin completing the square. Divide all terms by -676 the coefficient of the squared term: Divide each side by '-676'. 0.06360946746 + -1.062130178x + x2 = 0 Move the constant term to the right: Add '-0.06360946746' to each side of the equation. 0.06360946746 + -1.062130178x + -0.06360946746 + x2 = 0 + -0.06360946746 Reorder the terms: 0.06360946746 + -0.06360946746 + -1.062130178x + x2 = 0 + -0.06360946746 Combine like terms: 0.06360946746 + -0.06360946746 = 0.00000000000 0.00000000000 + -1.062130178x + x2 = 0 + -0.06360946746 -1.062130178x + x2 = 0 + -0.06360946746 Combine like terms: 0 + -0.06360946746 = -0.06360946746 -1.062130178x + x2 = -0.06360946746 The x term is -1.062130178x. Take half its coefficient (-0.531065089). Square it (0.2820301288) and add it to both sides. Add '0.2820301288' to each side of the equation. -1.062130178x + 0.2820301288 + x2 = -0.06360946746 + 0.2820301288 Reorder the terms: 0.2820301288 + -1.062130178x + x2 = -0.06360946746 + 0.2820301288 Combine like terms: -0.06360946746 + 0.2820301288 = 0.21842066134 0.2820301288 + -1.062130178x + x2 = 0.21842066134 Factor a perfect square on the left side: (x + -0.531065089)(x + -0.531065089) = 0.21842066134 Calculate the square root of the right side: 0.467354963 Break this problem into two subproblems by setting (x + -0.531065089) equal to 0.467354963 and -0.467354963.Subproblem 1
x + -0.531065089 = 0.467354963 Simplifying x + -0.531065089 = 0.467354963 Reorder the terms: -0.531065089 + x = 0.467354963 Solving -0.531065089 + x = 0.467354963 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.531065089' to each side of the equation. -0.531065089 + 0.531065089 + x = 0.467354963 + 0.531065089 Combine like terms: -0.531065089 + 0.531065089 = 0.000000000 0.000000000 + x = 0.467354963 + 0.531065089 x = 0.467354963 + 0.531065089 Combine like terms: 0.467354963 + 0.531065089 = 0.998420052 x = 0.998420052 Simplifying x = 0.998420052Subproblem 2
x + -0.531065089 = -0.467354963 Simplifying x + -0.531065089 = -0.467354963 Reorder the terms: -0.531065089 + x = -0.467354963 Solving -0.531065089 + x = -0.467354963 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.531065089' to each side of the equation. -0.531065089 + 0.531065089 + x = -0.467354963 + 0.531065089 Combine like terms: -0.531065089 + 0.531065089 = 0.000000000 0.000000000 + x = -0.467354963 + 0.531065089 x = -0.467354963 + 0.531065089 Combine like terms: -0.467354963 + 0.531065089 = 0.063710126 x = 0.063710126 Simplifying x = 0.063710126Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.998420052, 0.063710126}
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