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Simplifying (13x)(x + -3) = x + 32 Remove parenthesis around (13x) 13x(x + -3) = x + 32 Reorder the terms: 13x(-3 + x) = x + 32 (-3 * 13x + x * 13x) = x + 32 (-39x + 13x2) = x + 32 Reorder the terms: -39x + 13x2 = 32 + x Solving -39x + 13x2 = 32 + x Solving for variable 'x'. Reorder the terms: -32 + -39x + -1x + 13x2 = 32 + x + -32 + -1x Combine like terms: -39x + -1x = -40x -32 + -40x + 13x2 = 32 + x + -32 + -1x Reorder the terms: -32 + -40x + 13x2 = 32 + -32 + x + -1x Combine like terms: 32 + -32 = 0 -32 + -40x + 13x2 = 0 + x + -1x -32 + -40x + 13x2 = x + -1x Combine like terms: x + -1x = 0 -32 + -40x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -2.461538462 + -3.076923077x + x2 = 0 Move the constant term to the right: Add '2.461538462' to each side of the equation. -2.461538462 + -3.076923077x + 2.461538462 + x2 = 0 + 2.461538462 Reorder the terms: -2.461538462 + 2.461538462 + -3.076923077x + x2 = 0 + 2.461538462 Combine like terms: -2.461538462 + 2.461538462 = 0.000000000 0.000000000 + -3.076923077x + x2 = 0 + 2.461538462 -3.076923077x + x2 = 0 + 2.461538462 Combine like terms: 0 + 2.461538462 = 2.461538462 -3.076923077x + x2 = 2.461538462 The x term is -3.076923077x. Take half its coefficient (-1.538461539). Square it (2.366863907) and add it to both sides. Add '2.366863907' to each side of the equation. -3.076923077x + 2.366863907 + x2 = 2.461538462 + 2.366863907 Reorder the terms: 2.366863907 + -3.076923077x + x2 = 2.461538462 + 2.366863907 Combine like terms: 2.461538462 + 2.366863907 = 4.828402369 2.366863907 + -3.076923077x + x2 = 4.828402369 Factor a perfect square on the left side: (x + -1.538461539)(x + -1.538461539) = 4.828402369 Calculate the square root of the right side: 2.197362594 Break this problem into two subproblems by setting (x + -1.538461539) equal to 2.197362594 and -2.197362594.Subproblem 1
x + -1.538461539 = 2.197362594 Simplifying x + -1.538461539 = 2.197362594 Reorder the terms: -1.538461539 + x = 2.197362594 Solving -1.538461539 + x = 2.197362594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.538461539' to each side of the equation. -1.538461539 + 1.538461539 + x = 2.197362594 + 1.538461539 Combine like terms: -1.538461539 + 1.538461539 = 0.000000000 0.000000000 + x = 2.197362594 + 1.538461539 x = 2.197362594 + 1.538461539 Combine like terms: 2.197362594 + 1.538461539 = 3.735824133 x = 3.735824133 Simplifying x = 3.735824133Subproblem 2
x + -1.538461539 = -2.197362594 Simplifying x + -1.538461539 = -2.197362594 Reorder the terms: -1.538461539 + x = -2.197362594 Solving -1.538461539 + x = -2.197362594 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.538461539' to each side of the equation. -1.538461539 + 1.538461539 + x = -2.197362594 + 1.538461539 Combine like terms: -1.538461539 + 1.538461539 = 0.000000000 0.000000000 + x = -2.197362594 + 1.538461539 x = -2.197362594 + 1.538461539 Combine like terms: -2.197362594 + 1.538461539 = -0.658901055 x = -0.658901055 Simplifying x = -0.658901055Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.735824133, -0.658901055}
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