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Simplifying (3x + 2) * 5x = 90 Reorder the terms: (2 + 3x) * 5x = 90 Reorder the terms for easier multiplication: 5x(2 + 3x) = 90 (2 * 5x + 3x * 5x) = 90 (10x + 15x2) = 90 Solving 10x + 15x2 = 90 Solving for variable 'x'. Reorder the terms: -90 + 10x + 15x2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + 10x + 15x2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-18 + 2x + 3x2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-18 + 2x + 3x2)' equal to zero and attempt to solve: Simplifying -18 + 2x + 3x2 = 0 Solving -18 + 2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -6 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 0.6666666667x + 6 + x2 = 0 + 6 Reorder the terms: -6 + 6 + 0.6666666667x + x2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 0.6666666667x + x2 = 0 + 6 0.6666666667x + x2 = 0 + 6 Combine like terms: 0 + 6 = 6 0.6666666667x + x2 = 6 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 6 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 6 + 0.1111111112 Combine like terms: 6 + 0.1111111112 = 6.1111111112 0.1111111112 + 0.6666666667x + x2 = 6.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 6.1111111112 Calculate the square root of the right side: 2.472066162 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 2.472066162 and -2.472066162.Subproblem 1
x + 0.3333333334 = 2.472066162 Simplifying x + 0.3333333334 = 2.472066162 Reorder the terms: 0.3333333334 + x = 2.472066162 Solving 0.3333333334 + x = 2.472066162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 2.472066162 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 2.472066162 + -0.3333333334 x = 2.472066162 + -0.3333333334 Combine like terms: 2.472066162 + -0.3333333334 = 2.1387328286 x = 2.1387328286 Simplifying x = 2.1387328286Subproblem 2
x + 0.3333333334 = -2.472066162 Simplifying x + 0.3333333334 = -2.472066162 Reorder the terms: 0.3333333334 + x = -2.472066162 Solving 0.3333333334 + x = -2.472066162 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -2.472066162 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -2.472066162 + -0.3333333334 x = -2.472066162 + -0.3333333334 Combine like terms: -2.472066162 + -0.3333333334 = -2.8053994954 x = -2.8053994954 Simplifying x = -2.8053994954Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.1387328286, -2.8053994954}Solution
x = {2.1387328286, -2.8053994954}
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