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Simplifying 145 = (x + 1)(4x + 1) Reorder the terms: 145 = (1 + x)(4x + 1) Reorder the terms: 145 = (1 + x)(1 + 4x) Multiply (1 + x) * (1 + 4x) 145 = (1(1 + 4x) + x(1 + 4x)) 145 = ((1 * 1 + 4x * 1) + x(1 + 4x)) 145 = ((1 + 4x) + x(1 + 4x)) 145 = (1 + 4x + (1 * x + 4x * x)) 145 = (1 + 4x + (1x + 4x2)) Combine like terms: 4x + 1x = 5x 145 = (1 + 5x + 4x2) Solving 145 = 1 + 5x + 4x2 Solving for variable 'x'. Combine like terms: 145 + -1 = 144 144 + -5x + -4x2 = 1 + 5x + 4x2 + -1 + -5x + -4x2 Reorder the terms: 144 + -5x + -4x2 = 1 + -1 + 5x + -5x + 4x2 + -4x2 Combine like terms: 1 + -1 = 0 144 + -5x + -4x2 = 0 + 5x + -5x + 4x2 + -4x2 144 + -5x + -4x2 = 5x + -5x + 4x2 + -4x2 Combine like terms: 5x + -5x = 0 144 + -5x + -4x2 = 0 + 4x2 + -4x2 144 + -5x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 144 + -5x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -36 + 1.25x + x2 = 0 Move the constant term to the right: Add '36' to each side of the equation. -36 + 1.25x + 36 + x2 = 0 + 36 Reorder the terms: -36 + 36 + 1.25x + x2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + 1.25x + x2 = 0 + 36 1.25x + x2 = 0 + 36 Combine like terms: 0 + 36 = 36 1.25x + x2 = 36 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 36 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 36 + 0.390625 Combine like terms: 36 + 0.390625 = 36.390625 0.390625 + 1.25x + x2 = 36.390625 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 36.390625 Calculate the square root of the right side: 6.032464256 Break this problem into two subproblems by setting (x + 0.625) equal to 6.032464256 and -6.032464256.Subproblem 1
x + 0.625 = 6.032464256 Simplifying x + 0.625 = 6.032464256 Reorder the terms: 0.625 + x = 6.032464256 Solving 0.625 + x = 6.032464256 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 6.032464256 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 6.032464256 + -0.625 x = 6.032464256 + -0.625 Combine like terms: 6.032464256 + -0.625 = 5.407464256 x = 5.407464256 Simplifying x = 5.407464256Subproblem 2
x + 0.625 = -6.032464256 Simplifying x + 0.625 = -6.032464256 Reorder the terms: 0.625 + x = -6.032464256 Solving 0.625 + x = -6.032464256 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -6.032464256 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -6.032464256 + -0.625 x = -6.032464256 + -0.625 Combine like terms: -6.032464256 + -0.625 = -6.657464256 x = -6.657464256 Simplifying x = -6.657464256Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.407464256, -6.657464256}
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