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Simplifying 52 = 4x2 + 3x + 1 Reorder the terms: 52 = 1 + 3x + 4x2 Solving 52 = 1 + 3x + 4x2 Solving for variable 'x'. Combine like terms: 52 + -1 = 51 51 + -3x + -4x2 = 1 + 3x + 4x2 + -1 + -3x + -4x2 Reorder the terms: 51 + -3x + -4x2 = 1 + -1 + 3x + -3x + 4x2 + -4x2 Combine like terms: 1 + -1 = 0 51 + -3x + -4x2 = 0 + 3x + -3x + 4x2 + -4x2 51 + -3x + -4x2 = 3x + -3x + 4x2 + -4x2 Combine like terms: 3x + -3x = 0 51 + -3x + -4x2 = 0 + 4x2 + -4x2 51 + -3x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 51 + -3x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -12.75 + 0.75x + x2 = 0 Move the constant term to the right: Add '12.75' to each side of the equation. -12.75 + 0.75x + 12.75 + x2 = 0 + 12.75 Reorder the terms: -12.75 + 12.75 + 0.75x + x2 = 0 + 12.75 Combine like terms: -12.75 + 12.75 = 0.00 0.00 + 0.75x + x2 = 0 + 12.75 0.75x + x2 = 0 + 12.75 Combine like terms: 0 + 12.75 = 12.75 0.75x + x2 = 12.75 The x term is 0.75x. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75x + 0.140625 + x2 = 12.75 + 0.140625 Reorder the terms: 0.140625 + 0.75x + x2 = 12.75 + 0.140625 Combine like terms: 12.75 + 0.140625 = 12.890625 0.140625 + 0.75x + x2 = 12.890625 Factor a perfect square on the left side: (x + 0.375)(x + 0.375) = 12.890625 Calculate the square root of the right side: 3.590351654 Break this problem into two subproblems by setting (x + 0.375) equal to 3.590351654 and -3.590351654.Subproblem 1
x + 0.375 = 3.590351654 Simplifying x + 0.375 = 3.590351654 Reorder the terms: 0.375 + x = 3.590351654 Solving 0.375 + x = 3.590351654 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 3.590351654 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 3.590351654 + -0.375 x = 3.590351654 + -0.375 Combine like terms: 3.590351654 + -0.375 = 3.215351654 x = 3.215351654 Simplifying x = 3.215351654Subproblem 2
x + 0.375 = -3.590351654 Simplifying x + 0.375 = -3.590351654 Reorder the terms: 0.375 + x = -3.590351654 Solving 0.375 + x = -3.590351654 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -3.590351654 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -3.590351654 + -0.375 x = -3.590351654 + -0.375 Combine like terms: -3.590351654 + -0.375 = -3.965351654 x = -3.965351654 Simplifying x = -3.965351654Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.215351654, -3.965351654}
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