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Simplifying 3x + 4(4x + -5) * x = 3 Reorder the terms: 3x + 4(-5 + 4x) * x = 3 Reorder the terms for easier multiplication: 3x + 4x(-5 + 4x) = 3 3x + (-5 * 4x + 4x * 4x) = 3 3x + (-20x + 16x2) = 3 Combine like terms: 3x + -20x = -17x -17x + 16x2 = 3 Solving -17x + 16x2 = 3 Solving for variable 'x'. Reorder the terms: -3 + -17x + 16x2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -17x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.1875 + -1.0625x + x2 = 0 Move the constant term to the right: Add '0.1875' to each side of the equation. -0.1875 + -1.0625x + 0.1875 + x2 = 0 + 0.1875 Reorder the terms: -0.1875 + 0.1875 + -1.0625x + x2 = 0 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + -1.0625x + x2 = 0 + 0.1875 -1.0625x + x2 = 0 + 0.1875 Combine like terms: 0 + 0.1875 = 0.1875 -1.0625x + x2 = 0.1875 The x term is -1.0625x. Take half its coefficient (-0.53125). Square it (0.2822265625) and add it to both sides. Add '0.2822265625' to each side of the equation. -1.0625x + 0.2822265625 + x2 = 0.1875 + 0.2822265625 Reorder the terms: 0.2822265625 + -1.0625x + x2 = 0.1875 + 0.2822265625 Combine like terms: 0.1875 + 0.2822265625 = 0.4697265625 0.2822265625 + -1.0625x + x2 = 0.4697265625 Factor a perfect square on the left side: (x + -0.53125)(x + -0.53125) = 0.4697265625 Calculate the square root of the right side: 0.685366006 Break this problem into two subproblems by setting (x + -0.53125) equal to 0.685366006 and -0.685366006.Subproblem 1
x + -0.53125 = 0.685366006 Simplifying x + -0.53125 = 0.685366006 Reorder the terms: -0.53125 + x = 0.685366006 Solving -0.53125 + x = 0.685366006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.53125' to each side of the equation. -0.53125 + 0.53125 + x = 0.685366006 + 0.53125 Combine like terms: -0.53125 + 0.53125 = 0.00000 0.00000 + x = 0.685366006 + 0.53125 x = 0.685366006 + 0.53125 Combine like terms: 0.685366006 + 0.53125 = 1.216616006 x = 1.216616006 Simplifying x = 1.216616006Subproblem 2
x + -0.53125 = -0.685366006 Simplifying x + -0.53125 = -0.685366006 Reorder the terms: -0.53125 + x = -0.685366006 Solving -0.53125 + x = -0.685366006 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.53125' to each side of the equation. -0.53125 + 0.53125 + x = -0.685366006 + 0.53125 Combine like terms: -0.53125 + 0.53125 = 0.00000 0.00000 + x = -0.685366006 + 0.53125 x = -0.685366006 + 0.53125 Combine like terms: -0.685366006 + 0.53125 = -0.154116006 x = -0.154116006 Simplifying x = -0.154116006Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.216616006, -0.154116006}
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