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Simplifying 6x(3x + 10) = 14 Reorder the terms: 6x(10 + 3x) = 14 (10 * 6x + 3x * 6x) = 14 (60x + 18x2) = 14 Solving 60x + 18x2 = 14 Solving for variable 'x'. Reorder the terms: -14 + 60x + 18x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 60x + 18x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 30x + 9x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 30x + 9x2)' equal to zero and attempt to solve: Simplifying -7 + 30x + 9x2 = 0 Solving -7 + 30x + 9x2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.7777777778 + 3.333333333x + x2 = 0 Move the constant term to the right: Add '0.7777777778' to each side of the equation. -0.7777777778 + 3.333333333x + 0.7777777778 + x2 = 0 + 0.7777777778 Reorder the terms: -0.7777777778 + 0.7777777778 + 3.333333333x + x2 = 0 + 0.7777777778 Combine like terms: -0.7777777778 + 0.7777777778 = 0.0000000000 0.0000000000 + 3.333333333x + x2 = 0 + 0.7777777778 3.333333333x + x2 = 0 + 0.7777777778 Combine like terms: 0 + 0.7777777778 = 0.7777777778 3.333333333x + x2 = 0.7777777778 The x term is 3.333333333x. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333x + 2.777777779 + x2 = 0.7777777778 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333x + x2 = 0.7777777778 + 2.777777779 Combine like terms: 0.7777777778 + 2.777777779 = 3.5555555568 2.777777779 + 3.333333333x + x2 = 3.5555555568 Factor a perfect square on the left side: (x + 1.666666667)(x + 1.666666667) = 3.5555555568 Calculate the square root of the right side: 1.885618083 Break this problem into two subproblems by setting (x + 1.666666667) equal to 1.885618083 and -1.885618083.Subproblem 1
x + 1.666666667 = 1.885618083 Simplifying x + 1.666666667 = 1.885618083 Reorder the terms: 1.666666667 + x = 1.885618083 Solving 1.666666667 + x = 1.885618083 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = 1.885618083 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = 1.885618083 + -1.666666667 x = 1.885618083 + -1.666666667 Combine like terms: 1.885618083 + -1.666666667 = 0.218951416 x = 0.218951416 Simplifying x = 0.218951416Subproblem 2
x + 1.666666667 = -1.885618083 Simplifying x + 1.666666667 = -1.885618083 Reorder the terms: 1.666666667 + x = -1.885618083 Solving 1.666666667 + x = -1.885618083 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + x = -1.885618083 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + x = -1.885618083 + -1.666666667 x = -1.885618083 + -1.666666667 Combine like terms: -1.885618083 + -1.666666667 = -3.55228475 x = -3.55228475 Simplifying x = -3.55228475Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.218951416, -3.55228475}Solution
x = {0.218951416, -3.55228475}
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