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Simplifying 175 = (4x + 1)(4x) Reorder the terms: 175 = (1 + 4x)(4x) Remove parenthesis around (4x) 175 = (1 + 4x) * 4x Reorder the terms for easier multiplication: 175 = 4x(1 + 4x) 175 = (1 * 4x + 4x * 4x) 175 = (4x + 16x2) Solving 175 = 4x + 16x2 Solving for variable 'x'. Reorder the terms: 175 + -4x + -16x2 = 4x + -4x + 16x2 + -16x2 Combine like terms: 4x + -4x = 0 175 + -4x + -16x2 = 0 + 16x2 + -16x2 175 + -4x + -16x2 = 16x2 + -16x2 Combine like terms: 16x2 + -16x2 = 0 175 + -4x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -10.9375 + 0.25x + x2 = 0 Move the constant term to the right: Add '10.9375' to each side of the equation. -10.9375 + 0.25x + 10.9375 + x2 = 0 + 10.9375 Reorder the terms: -10.9375 + 10.9375 + 0.25x + x2 = 0 + 10.9375 Combine like terms: -10.9375 + 10.9375 = 0.0000 0.0000 + 0.25x + x2 = 0 + 10.9375 0.25x + x2 = 0 + 10.9375 Combine like terms: 0 + 10.9375 = 10.9375 0.25x + x2 = 10.9375 The x term is 0.25x. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25x + 0.015625 + x2 = 10.9375 + 0.015625 Reorder the terms: 0.015625 + 0.25x + x2 = 10.9375 + 0.015625 Combine like terms: 10.9375 + 0.015625 = 10.953125 0.015625 + 0.25x + x2 = 10.953125 Factor a perfect square on the left side: (x + 0.125)(x + 0.125) = 10.953125 Calculate the square root of the right side: 3.309550574 Break this problem into two subproblems by setting (x + 0.125) equal to 3.309550574 and -3.309550574.Subproblem 1
x + 0.125 = 3.309550574 Simplifying x + 0.125 = 3.309550574 Reorder the terms: 0.125 + x = 3.309550574 Solving 0.125 + x = 3.309550574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 3.309550574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 3.309550574 + -0.125 x = 3.309550574 + -0.125 Combine like terms: 3.309550574 + -0.125 = 3.184550574 x = 3.184550574 Simplifying x = 3.184550574Subproblem 2
x + 0.125 = -3.309550574 Simplifying x + 0.125 = -3.309550574 Reorder the terms: 0.125 + x = -3.309550574 Solving 0.125 + x = -3.309550574 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -3.309550574 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -3.309550574 + -0.125 x = -3.309550574 + -0.125 Combine like terms: -3.309550574 + -0.125 = -3.434550574 x = -3.434550574 Simplifying x = -3.434550574Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.184550574, -3.434550574}
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