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Simplifying 3(2x + 3) * 7x + 27 + 13x = 32 Reorder the terms: 3(3 + 2x) * 7x + 27 + 13x = 32 Reorder the terms for easier multiplication: 3 * 7x(3 + 2x) + 27 + 13x = 32 Multiply 3 * 7 21x(3 + 2x) + 27 + 13x = 32 (3 * 21x + 2x * 21x) + 27 + 13x = 32 (63x + 42x2) + 27 + 13x = 32 Reorder the terms: 27 + 63x + 13x + 42x2 = 32 Combine like terms: 63x + 13x = 76x 27 + 76x + 42x2 = 32 Solving 27 + 76x + 42x2 = 32 Solving for variable 'x'. Reorder the terms: 27 + -32 + 76x + 42x2 = 32 + -32 Combine like terms: 27 + -32 = -5 -5 + 76x + 42x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -5 + 76x + 42x2 = 0 Begin completing the square. Divide all terms by 42 the coefficient of the squared term: Divide each side by '42'. -0.119047619 + 1.80952381x + x2 = 0 Move the constant term to the right: Add '0.119047619' to each side of the equation. -0.119047619 + 1.80952381x + 0.119047619 + x2 = 0 + 0.119047619 Reorder the terms: -0.119047619 + 0.119047619 + 1.80952381x + x2 = 0 + 0.119047619 Combine like terms: -0.119047619 + 0.119047619 = 0.000000000 0.000000000 + 1.80952381x + x2 = 0 + 0.119047619 1.80952381x + x2 = 0 + 0.119047619 Combine like terms: 0 + 0.119047619 = 0.119047619 1.80952381x + x2 = 0.119047619 The x term is 1.80952381x. Take half its coefficient (0.904761905). Square it (0.8185941047) and add it to both sides. Add '0.8185941047' to each side of the equation. 1.80952381x + 0.8185941047 + x2 = 0.119047619 + 0.8185941047 Reorder the terms: 0.8185941047 + 1.80952381x + x2 = 0.119047619 + 0.8185941047 Combine like terms: 0.119047619 + 0.8185941047 = 0.9376417237 0.8185941047 + 1.80952381x + x2 = 0.9376417237 Factor a perfect square on the left side: (x + 0.904761905)(x + 0.904761905) = 0.9376417237 Calculate the square root of the right side: 0.96831902 Break this problem into two subproblems by setting (x + 0.904761905) equal to 0.96831902 and -0.96831902.Subproblem 1
x + 0.904761905 = 0.96831902 Simplifying x + 0.904761905 = 0.96831902 Reorder the terms: 0.904761905 + x = 0.96831902 Solving 0.904761905 + x = 0.96831902 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.904761905' to each side of the equation. 0.904761905 + -0.904761905 + x = 0.96831902 + -0.904761905 Combine like terms: 0.904761905 + -0.904761905 = 0.000000000 0.000000000 + x = 0.96831902 + -0.904761905 x = 0.96831902 + -0.904761905 Combine like terms: 0.96831902 + -0.904761905 = 0.063557115 x = 0.063557115 Simplifying x = 0.063557115Subproblem 2
x + 0.904761905 = -0.96831902 Simplifying x + 0.904761905 = -0.96831902 Reorder the terms: 0.904761905 + x = -0.96831902 Solving 0.904761905 + x = -0.96831902 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.904761905' to each side of the equation. 0.904761905 + -0.904761905 + x = -0.96831902 + -0.904761905 Combine like terms: 0.904761905 + -0.904761905 = 0.000000000 0.000000000 + x = -0.96831902 + -0.904761905 x = -0.96831902 + -0.904761905 Combine like terms: -0.96831902 + -0.904761905 = -1.873080925 x = -1.873080925 Simplifying x = -1.873080925Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.063557115, -1.873080925}
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