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Simplifying 2x + 2(5 + 2x) * 6x + 10 = 250 Reorder the terms for easier multiplication: 2x + 2 * 6x(5 + 2x) + 10 = 250 Multiply 2 * 6 2x + 12x(5 + 2x) + 10 = 250 2x + (5 * 12x + 2x * 12x) + 10 = 250 2x + (60x + 24x2) + 10 = 250 Reorder the terms: 10 + 2x + 60x + 24x2 = 250 Combine like terms: 2x + 60x = 62x 10 + 62x + 24x2 = 250 Solving 10 + 62x + 24x2 = 250 Solving for variable 'x'. Reorder the terms: 10 + -250 + 62x + 24x2 = 250 + -250 Combine like terms: 10 + -250 = -240 -240 + 62x + 24x2 = 250 + -250 Combine like terms: 250 + -250 = 0 -240 + 62x + 24x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-120 + 31x + 12x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-120 + 31x + 12x2)' equal to zero and attempt to solve: Simplifying -120 + 31x + 12x2 = 0 Solving -120 + 31x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -10 + 2.583333333x + x2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 2.583333333x + 10 + x2 = 0 + 10 Reorder the terms: -10 + 10 + 2.583333333x + x2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 2.583333333x + x2 = 0 + 10 2.583333333x + x2 = 0 + 10 Combine like terms: 0 + 10 = 10 2.583333333x + x2 = 10 The x term is 2.583333333x. Take half its coefficient (1.291666667). Square it (1.668402779) and add it to both sides. Add '1.668402779' to each side of the equation. 2.583333333x + 1.668402779 + x2 = 10 + 1.668402779 Reorder the terms: 1.668402779 + 2.583333333x + x2 = 10 + 1.668402779 Combine like terms: 10 + 1.668402779 = 11.668402779 1.668402779 + 2.583333333x + x2 = 11.668402779 Factor a perfect square on the left side: (x + 1.291666667)(x + 1.291666667) = 11.668402779 Calculate the square root of the right side: 3.415904387 Break this problem into two subproblems by setting (x + 1.291666667) equal to 3.415904387 and -3.415904387.Subproblem 1
x + 1.291666667 = 3.415904387 Simplifying x + 1.291666667 = 3.415904387 Reorder the terms: 1.291666667 + x = 3.415904387 Solving 1.291666667 + x = 3.415904387 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.291666667' to each side of the equation. 1.291666667 + -1.291666667 + x = 3.415904387 + -1.291666667 Combine like terms: 1.291666667 + -1.291666667 = 0.000000000 0.000000000 + x = 3.415904387 + -1.291666667 x = 3.415904387 + -1.291666667 Combine like terms: 3.415904387 + -1.291666667 = 2.12423772 x = 2.12423772 Simplifying x = 2.12423772Subproblem 2
x + 1.291666667 = -3.415904387 Simplifying x + 1.291666667 = -3.415904387 Reorder the terms: 1.291666667 + x = -3.415904387 Solving 1.291666667 + x = -3.415904387 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.291666667' to each side of the equation. 1.291666667 + -1.291666667 + x = -3.415904387 + -1.291666667 Combine like terms: 1.291666667 + -1.291666667 = 0.000000000 0.000000000 + x = -3.415904387 + -1.291666667 x = -3.415904387 + -1.291666667 Combine like terms: -3.415904387 + -1.291666667 = -4.707571054 x = -4.707571054 Simplifying x = -4.707571054Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.12423772, -4.707571054}Solution
x = {2.12423772, -4.707571054}
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