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Simplifying 3 + 2x + 4x(4x) = 140 Remove parenthesis around (4x) 3 + 2x + 4x * 4x = 140 Reorder the terms for easier multiplication: 3 + 2x + 4 * 4x * x = 140 Multiply 4 * 4 3 + 2x + 16x * x = 140 Multiply x * x 3 + 2x + 16x2 = 140 Solving 3 + 2x + 16x2 = 140 Solving for variable 'x'. Reorder the terms: 3 + -140 + 2x + 16x2 = 140 + -140 Combine like terms: 3 + -140 = -137 -137 + 2x + 16x2 = 140 + -140 Combine like terms: 140 + -140 = 0 -137 + 2x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -8.5625 + 0.125x + x2 = 0 Move the constant term to the right: Add '8.5625' to each side of the equation. -8.5625 + 0.125x + 8.5625 + x2 = 0 + 8.5625 Reorder the terms: -8.5625 + 8.5625 + 0.125x + x2 = 0 + 8.5625 Combine like terms: -8.5625 + 8.5625 = 0.0000 0.0000 + 0.125x + x2 = 0 + 8.5625 0.125x + x2 = 0 + 8.5625 Combine like terms: 0 + 8.5625 = 8.5625 0.125x + x2 = 8.5625 The x term is 0.125x. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125x + 0.00390625 + x2 = 8.5625 + 0.00390625 Reorder the terms: 0.00390625 + 0.125x + x2 = 8.5625 + 0.00390625 Combine like terms: 8.5625 + 0.00390625 = 8.56640625 0.00390625 + 0.125x + x2 = 8.56640625 Factor a perfect square on the left side: (x + 0.0625)(x + 0.0625) = 8.56640625 Calculate the square root of the right side: 2.926842368 Break this problem into two subproblems by setting (x + 0.0625) equal to 2.926842368 and -2.926842368.Subproblem 1
x + 0.0625 = 2.926842368 Simplifying x + 0.0625 = 2.926842368 Reorder the terms: 0.0625 + x = 2.926842368 Solving 0.0625 + x = 2.926842368 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + x = 2.926842368 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = 2.926842368 + -0.0625 x = 2.926842368 + -0.0625 Combine like terms: 2.926842368 + -0.0625 = 2.864342368 x = 2.864342368 Simplifying x = 2.864342368Subproblem 2
x + 0.0625 = -2.926842368 Simplifying x + 0.0625 = -2.926842368 Reorder the terms: 0.0625 + x = -2.926842368 Solving 0.0625 + x = -2.926842368 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + x = -2.926842368 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = -2.926842368 + -0.0625 x = -2.926842368 + -0.0625 Combine like terms: -2.926842368 + -0.0625 = -2.989342368 x = -2.989342368 Simplifying x = -2.989342368Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.864342368, -2.989342368}
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