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Simplifying 4x2 + 32x + 23 = x + 7 Reorder the terms: 23 + 32x + 4x2 = x + 7 Reorder the terms: 23 + 32x + 4x2 = 7 + x Solving 23 + 32x + 4x2 = 7 + x Solving for variable 'x'. Reorder the terms: 23 + -7 + 32x + -1x + 4x2 = 7 + x + -7 + -1x Combine like terms: 23 + -7 = 16 16 + 32x + -1x + 4x2 = 7 + x + -7 + -1x Combine like terms: 32x + -1x = 31x 16 + 31x + 4x2 = 7 + x + -7 + -1x Reorder the terms: 16 + 31x + 4x2 = 7 + -7 + x + -1x Combine like terms: 7 + -7 = 0 16 + 31x + 4x2 = 0 + x + -1x 16 + 31x + 4x2 = x + -1x Combine like terms: x + -1x = 0 16 + 31x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 4 + 7.75x + x2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + 7.75x + -4 + x2 = 0 + -4 Reorder the terms: 4 + -4 + 7.75x + x2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + 7.75x + x2 = 0 + -4 7.75x + x2 = 0 + -4 Combine like terms: 0 + -4 = -4 7.75x + x2 = -4 The x term is 7.75x. Take half its coefficient (3.875). Square it (15.015625) and add it to both sides. Add '15.015625' to each side of the equation. 7.75x + 15.015625 + x2 = -4 + 15.015625 Reorder the terms: 15.015625 + 7.75x + x2 = -4 + 15.015625 Combine like terms: -4 + 15.015625 = 11.015625 15.015625 + 7.75x + x2 = 11.015625 Factor a perfect square on the left side: (x + 3.875)(x + 3.875) = 11.015625 Calculate the square root of the right side: 3.318979512 Break this problem into two subproblems by setting (x + 3.875) equal to 3.318979512 and -3.318979512.Subproblem 1
x + 3.875 = 3.318979512 Simplifying x + 3.875 = 3.318979512 Reorder the terms: 3.875 + x = 3.318979512 Solving 3.875 + x = 3.318979512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = 3.318979512 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = 3.318979512 + -3.875 x = 3.318979512 + -3.875 Combine like terms: 3.318979512 + -3.875 = -0.556020488 x = -0.556020488 Simplifying x = -0.556020488Subproblem 2
x + 3.875 = -3.318979512 Simplifying x + 3.875 = -3.318979512 Reorder the terms: 3.875 + x = -3.318979512 Solving 3.875 + x = -3.318979512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.875' to each side of the equation. 3.875 + -3.875 + x = -3.318979512 + -3.875 Combine like terms: 3.875 + -3.875 = 0.000 0.000 + x = -3.318979512 + -3.875 x = -3.318979512 + -3.875 Combine like terms: -3.318979512 + -3.875 = -7.193979512 x = -7.193979512 Simplifying x = -7.193979512Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.556020488, -7.193979512}
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