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Simplifying 12x2 + 10x = 16 Reorder the terms: 10x + 12x2 = 16 Solving 10x + 12x2 = 16 Solving for variable 'x'. Reorder the terms: -16 + 10x + 12x2 = 16 + -16 Combine like terms: 16 + -16 = 0 -16 + 10x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-8 + 5x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-8 + 5x + 6x2)' equal to zero and attempt to solve: Simplifying -8 + 5x + 6x2 = 0 Solving -8 + 5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 0.8333333333x + x2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 0.8333333333x + 1.333333333 + x2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 0.8333333333x + x2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 0.8333333333x + x2 = 0 + 1.333333333 0.8333333333x + x2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 0.8333333333x + x2 = 1.333333333 The x term is 0.8333333333x. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333x + 0.1736111111 + x2 = 1.333333333 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333x + x2 = 1.333333333 + 0.1736111111 Combine like terms: 1.333333333 + 0.1736111111 = 1.5069444441 0.1736111111 + 0.8333333333x + x2 = 1.5069444441 Factor a perfect square on the left side: (x + 0.4166666667)(x + 0.4166666667) = 1.5069444441 Calculate the square root of the right side: 1.227576655 Break this problem into two subproblems by setting (x + 0.4166666667) equal to 1.227576655 and -1.227576655.Subproblem 1
x + 0.4166666667 = 1.227576655 Simplifying x + 0.4166666667 = 1.227576655 Reorder the terms: 0.4166666667 + x = 1.227576655 Solving 0.4166666667 + x = 1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = 1.227576655 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = 1.227576655 + -0.4166666667 x = 1.227576655 + -0.4166666667 Combine like terms: 1.227576655 + -0.4166666667 = 0.8109099883 x = 0.8109099883 Simplifying x = 0.8109099883Subproblem 2
x + 0.4166666667 = -1.227576655 Simplifying x + 0.4166666667 = -1.227576655 Reorder the terms: 0.4166666667 + x = -1.227576655 Solving 0.4166666667 + x = -1.227576655 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + x = -1.227576655 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + x = -1.227576655 + -0.4166666667 x = -1.227576655 + -0.4166666667 Combine like terms: -1.227576655 + -0.4166666667 = -1.6442433217 x = -1.6442433217 Simplifying x = -1.6442433217Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.8109099883, -1.6442433217}Solution
x = {0.8109099883, -1.6442433217}
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