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Simplifying 3a + 2a * 7a + 8654 = 32654 Reorder the terms for easier multiplication: 3a + 2 * 7a * a + 8654 = 32654 Multiply 2 * 7 3a + 14a * a + 8654 = 32654 Multiply a * a 3a + 14a2 + 8654 = 32654 Reorder the terms: 8654 + 3a + 14a2 = 32654 Solving 8654 + 3a + 14a2 = 32654 Solving for variable 'a'. Reorder the terms: 8654 + -32654 + 3a + 14a2 = 32654 + -32654 Combine like terms: 8654 + -32654 = -24000 -24000 + 3a + 14a2 = 32654 + -32654 Combine like terms: 32654 + -32654 = 0 -24000 + 3a + 14a2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1714.285714 + 0.2142857143a + a2 = 0 Move the constant term to the right: Add '1714.285714' to each side of the equation. -1714.285714 + 0.2142857143a + 1714.285714 + a2 = 0 + 1714.285714 Reorder the terms: -1714.285714 + 1714.285714 + 0.2142857143a + a2 = 0 + 1714.285714 Combine like terms: -1714.285714 + 1714.285714 = 0.000000 0.000000 + 0.2142857143a + a2 = 0 + 1714.285714 0.2142857143a + a2 = 0 + 1714.285714 Combine like terms: 0 + 1714.285714 = 1714.285714 0.2142857143a + a2 = 1714.285714 The a term is 0.2142857143a. Take half its coefficient (0.1071428572). Square it (0.01147959185) and add it to both sides. Add '0.01147959185' to each side of the equation. 0.2142857143a + 0.01147959185 + a2 = 1714.285714 + 0.01147959185 Reorder the terms: 0.01147959185 + 0.2142857143a + a2 = 1714.285714 + 0.01147959185 Combine like terms: 1714.285714 + 0.01147959185 = 1714.29719359185 0.01147959185 + 0.2142857143a + a2 = 1714.29719359185 Factor a perfect square on the left side: (a + 0.1071428572)(a + 0.1071428572) = 1714.29719359185 Calculate the square root of the right side: 41.404072186 Break this problem into two subproblems by setting (a + 0.1071428572) equal to 41.404072186 and -41.404072186.Subproblem 1
a + 0.1071428572 = 41.404072186 Simplifying a + 0.1071428572 = 41.404072186 Reorder the terms: 0.1071428572 + a = 41.404072186 Solving 0.1071428572 + a = 41.404072186 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + a = 41.404072186 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + a = 41.404072186 + -0.1071428572 a = 41.404072186 + -0.1071428572 Combine like terms: 41.404072186 + -0.1071428572 = 41.2969293288 a = 41.2969293288 Simplifying a = 41.2969293288Subproblem 2
a + 0.1071428572 = -41.404072186 Simplifying a + 0.1071428572 = -41.404072186 Reorder the terms: 0.1071428572 + a = -41.404072186 Solving 0.1071428572 + a = -41.404072186 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + a = -41.404072186 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + a = -41.404072186 + -0.1071428572 a = -41.404072186 + -0.1071428572 Combine like terms: -41.404072186 + -0.1071428572 = -41.5112150432 a = -41.5112150432 Simplifying a = -41.5112150432Solution
The solution to the problem is based on the solutions from the subproblems. a = {41.2969293288, -41.5112150432}
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