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Simplifying 40x + 4x(1.5x) = 345 Remove parenthesis around (1.5x) 40x + 4x * 1.5x = 345 Reorder the terms for easier multiplication: 40x + 4 * 1.5x * x = 345 Multiply 4 * 1.5 40x + 6x * x = 345 Multiply x * x 40x + 6x2 = 345 Solving 40x + 6x2 = 345 Solving for variable 'x'. Reorder the terms: -345 + 40x + 6x2 = 345 + -345 Combine like terms: 345 + -345 = 0 -345 + 40x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -57.5 + 6.666666667x + x2 = 0 Move the constant term to the right: Add '57.5' to each side of the equation. -57.5 + 6.666666667x + 57.5 + x2 = 0 + 57.5 Reorder the terms: -57.5 + 57.5 + 6.666666667x + x2 = 0 + 57.5 Combine like terms: -57.5 + 57.5 = 0.0 0.0 + 6.666666667x + x2 = 0 + 57.5 6.666666667x + x2 = 0 + 57.5 Combine like terms: 0 + 57.5 = 57.5 6.666666667x + x2 = 57.5 The x term is 6.666666667x. Take half its coefficient (3.333333334). Square it (11.11111112) and add it to both sides. Add '11.11111112' to each side of the equation. 6.666666667x + 11.11111112 + x2 = 57.5 + 11.11111112 Reorder the terms: 11.11111112 + 6.666666667x + x2 = 57.5 + 11.11111112 Combine like terms: 57.5 + 11.11111112 = 68.61111112 11.11111112 + 6.666666667x + x2 = 68.61111112 Factor a perfect square on the left side: (x + 3.333333334)(x + 3.333333334) = 68.61111112 Calculate the square root of the right side: 8.283182427 Break this problem into two subproblems by setting (x + 3.333333334) equal to 8.283182427 and -8.283182427.Subproblem 1
x + 3.333333334 = 8.283182427 Simplifying x + 3.333333334 = 8.283182427 Reorder the terms: 3.333333334 + x = 8.283182427 Solving 3.333333334 + x = 8.283182427 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = 8.283182427 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = 8.283182427 + -3.333333334 x = 8.283182427 + -3.333333334 Combine like terms: 8.283182427 + -3.333333334 = 4.949849093 x = 4.949849093 Simplifying x = 4.949849093Subproblem 2
x + 3.333333334 = -8.283182427 Simplifying x + 3.333333334 = -8.283182427 Reorder the terms: 3.333333334 + x = -8.283182427 Solving 3.333333334 + x = -8.283182427 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = -8.283182427 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = -8.283182427 + -3.333333334 x = -8.283182427 + -3.333333334 Combine like terms: -8.283182427 + -3.333333334 = -11.616515761 x = -11.616515761 Simplifying x = -11.616515761Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.949849093, -11.616515761}
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