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Simplifying 4(12x + 5) + -6x(8x + 20 + -6x + 12) = 1 + -1(x)(x) + -1(5)(5) Reorder the terms: 4(5 + 12x) + -6x(8x + 20 + -6x + 12) = 1 + -1(x)(x) + -1(5)(5) (5 * 4 + 12x * 4) + -6x(8x + 20 + -6x + 12) = 1 + -1(x)(x) + -1(5)(5) (20 + 48x) + -6x(8x + 20 + -6x + 12) = 1 + -1(x)(x) + -1(5)(5) Reorder the terms: 20 + 48x + -6x(20 + 12 + 8x + -6x) = 1 + -1(x)(x) + -1(5)(5) Combine like terms: 20 + 12 = 32 20 + 48x + -6x(32 + 8x + -6x) = 1 + -1(x)(x) + -1(5)(5) Combine like terms: 8x + -6x = 2x 20 + 48x + -6x(32 + 2x) = 1 + -1(x)(x) + -1(5)(5) 20 + 48x + (32 * -6x + 2x * -6x) = 1 + -1(x)(x) + -1(5)(5) 20 + 48x + (-192x + -12x2) = 1 + -1(x)(x) + -1(5)(5) Combine like terms: 48x + -192x = -144x 20 + -144x + -12x2 = 1 + -1(x)(x) + -1(5)(5) Multiply x * x 20 + -144x + -12x2 = 1 + -1x2 + -1(5)(5) Multiply -1 * 5 20 + -144x + -12x2 = 1 + -1x2 + -5 * 5 Multiply -5 * 5 20 + -144x + -12x2 = 1 + -1x2 + -25 Reorder the terms: 20 + -144x + -12x2 = 1 + -25 + -1x2 Combine like terms: 1 + -25 = -24 20 + -144x + -12x2 = -24 + -1x2 Solving 20 + -144x + -12x2 = -24 + -1x2 Solving for variable 'x'. Reorder the terms: 20 + 24 + -144x + -12x2 + x2 = -24 + -1x2 + 24 + x2 Combine like terms: 20 + 24 = 44 44 + -144x + -12x2 + x2 = -24 + -1x2 + 24 + x2 Combine like terms: -12x2 + x2 = -11x2 44 + -144x + -11x2 = -24 + -1x2 + 24 + x2 Reorder the terms: 44 + -144x + -11x2 = -24 + 24 + -1x2 + x2 Combine like terms: -24 + 24 = 0 44 + -144x + -11x2 = 0 + -1x2 + x2 44 + -144x + -11x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 44 + -144x + -11x2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -4 + 13.09090909x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 13.09090909x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 13.09090909x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 13.09090909x + x2 = 0 + 4 13.09090909x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 13.09090909x + x2 = 4 The x term is 13.09090909x. Take half its coefficient (6.545454545). Square it (42.84297520) and add it to both sides. Add '42.84297520' to each side of the equation. 13.09090909x + 42.84297520 + x2 = 4 + 42.84297520 Reorder the terms: 42.84297520 + 13.09090909x + x2 = 4 + 42.84297520 Combine like terms: 4 + 42.84297520 = 46.8429752 42.84297520 + 13.09090909x + x2 = 46.8429752 Factor a perfect square on the left side: (x + 6.545454545)(x + 6.545454545) = 46.8429752 Calculate the square root of the right side: 6.844192809 Break this problem into two subproblems by setting (x + 6.545454545) equal to 6.844192809 and -6.844192809.Subproblem 1
x + 6.545454545 = 6.844192809 Simplifying x + 6.545454545 = 6.844192809 Reorder the terms: 6.545454545 + x = 6.844192809 Solving 6.545454545 + x = 6.844192809 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.545454545' to each side of the equation. 6.545454545 + -6.545454545 + x = 6.844192809 + -6.545454545 Combine like terms: 6.545454545 + -6.545454545 = 0.000000000 0.000000000 + x = 6.844192809 + -6.545454545 x = 6.844192809 + -6.545454545 Combine like terms: 6.844192809 + -6.545454545 = 0.298738264 x = 0.298738264 Simplifying x = 0.298738264Subproblem 2
x + 6.545454545 = -6.844192809 Simplifying x + 6.545454545 = -6.844192809 Reorder the terms: 6.545454545 + x = -6.844192809 Solving 6.545454545 + x = -6.844192809 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.545454545' to each side of the equation. 6.545454545 + -6.545454545 + x = -6.844192809 + -6.545454545 Combine like terms: 6.545454545 + -6.545454545 = 0.000000000 0.000000000 + x = -6.844192809 + -6.545454545 x = -6.844192809 + -6.545454545 Combine like terms: -6.844192809 + -6.545454545 = -13.389647354 x = -13.389647354 Simplifying x = -13.389647354Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.298738264, -13.389647354}
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