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Simplifying 13x2 + 16x + 10 = 6x2 + x + 3 Reorder the terms: 10 + 16x + 13x2 = 6x2 + x + 3 Reorder the terms: 10 + 16x + 13x2 = 3 + x + 6x2 Solving 10 + 16x + 13x2 = 3 + x + 6x2 Solving for variable 'x'. Reorder the terms: 10 + -3 + 16x + -1x + 13x2 + -6x2 = 3 + x + 6x2 + -3 + -1x + -6x2 Combine like terms: 10 + -3 = 7 7 + 16x + -1x + 13x2 + -6x2 = 3 + x + 6x2 + -3 + -1x + -6x2 Combine like terms: 16x + -1x = 15x 7 + 15x + 13x2 + -6x2 = 3 + x + 6x2 + -3 + -1x + -6x2 Combine like terms: 13x2 + -6x2 = 7x2 7 + 15x + 7x2 = 3 + x + 6x2 + -3 + -1x + -6x2 Reorder the terms: 7 + 15x + 7x2 = 3 + -3 + x + -1x + 6x2 + -6x2 Combine like terms: 3 + -3 = 0 7 + 15x + 7x2 = 0 + x + -1x + 6x2 + -6x2 7 + 15x + 7x2 = x + -1x + 6x2 + -6x2 Combine like terms: x + -1x = 0 7 + 15x + 7x2 = 0 + 6x2 + -6x2 7 + 15x + 7x2 = 6x2 + -6x2 Combine like terms: 6x2 + -6x2 = 0 7 + 15x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1 + 2.142857143x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.142857143x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + 2.142857143x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.142857143x + x2 = 0 + -1 2.142857143x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.142857143x + x2 = -1 The x term is 2.142857143x. Take half its coefficient (1.071428572). Square it (1.147959185) and add it to both sides. Add '1.147959185' to each side of the equation. 2.142857143x + 1.147959185 + x2 = -1 + 1.147959185 Reorder the terms: 1.147959185 + 2.142857143x + x2 = -1 + 1.147959185 Combine like terms: -1 + 1.147959185 = 0.147959185 1.147959185 + 2.142857143x + x2 = 0.147959185 Factor a perfect square on the left side: (x + 1.071428572)(x + 1.071428572) = 0.147959185 Calculate the square root of the right side: 0.384654631 Break this problem into two subproblems by setting (x + 1.071428572) equal to 0.384654631 and -0.384654631.Subproblem 1
x + 1.071428572 = 0.384654631 Simplifying x + 1.071428572 = 0.384654631 Reorder the terms: 1.071428572 + x = 0.384654631 Solving 1.071428572 + x = 0.384654631 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + x = 0.384654631 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + x = 0.384654631 + -1.071428572 x = 0.384654631 + -1.071428572 Combine like terms: 0.384654631 + -1.071428572 = -0.686773941 x = -0.686773941 Simplifying x = -0.686773941Subproblem 2
x + 1.071428572 = -0.384654631 Simplifying x + 1.071428572 = -0.384654631 Reorder the terms: 1.071428572 + x = -0.384654631 Solving 1.071428572 + x = -0.384654631 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + x = -0.384654631 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + x = -0.384654631 + -1.071428572 x = -0.384654631 + -1.071428572 Combine like terms: -0.384654631 + -1.071428572 = -1.456083203 x = -1.456083203 Simplifying x = -1.456083203Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.686773941, -1.456083203}
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