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Simplifying (3x + 1) * 2 + -1(x + 5)(x + -5) = 4x(2x + -1) Reorder the terms: (1 + 3x) * 2 + -1(x + 5)(x + -5) = 4x(2x + -1) Reorder the terms for easier multiplication: 2(1 + 3x) + -1(x + 5)(x + -5) = 4x(2x + -1) (1 * 2 + 3x * 2) + -1(x + 5)(x + -5) = 4x(2x + -1) (2 + 6x) + -1(x + 5)(x + -5) = 4x(2x + -1) Reorder the terms: 2 + 6x + -1(5 + x)(x + -5) = 4x(2x + -1) Reorder the terms: 2 + 6x + -1(5 + x)(-5 + x) = 4x(2x + -1) Multiply (5 + x) * (-5 + x) 2 + 6x + -1(5(-5 + x) + x(-5 + x)) = 4x(2x + -1) 2 + 6x + -1((-5 * 5 + x * 5) + x(-5 + x)) = 4x(2x + -1) 2 + 6x + -1((-25 + 5x) + x(-5 + x)) = 4x(2x + -1) 2 + 6x + -1(-25 + 5x + (-5 * x + x * x)) = 4x(2x + -1) 2 + 6x + -1(-25 + 5x + (-5x + x2)) = 4x(2x + -1) Combine like terms: 5x + -5x = 0 2 + 6x + -1(-25 + 0 + x2) = 4x(2x + -1) 2 + 6x + -1(-25 + x2) = 4x(2x + -1) 2 + 6x + (-25 * -1 + x2 * -1) = 4x(2x + -1) 2 + 6x + (25 + -1x2) = 4x(2x + -1) Reorder the terms: 2 + 25 + 6x + -1x2 = 4x(2x + -1) Combine like terms: 2 + 25 = 27 27 + 6x + -1x2 = 4x(2x + -1) Reorder the terms: 27 + 6x + -1x2 = 4x(-1 + 2x) 27 + 6x + -1x2 = (-1 * 4x + 2x * 4x) 27 + 6x + -1x2 = (-4x + 8x2) Solving 27 + 6x + -1x2 = -4x + 8x2 Solving for variable 'x'. Reorder the terms: 27 + 6x + 4x + -1x2 + -8x2 = -4x + 8x2 + 4x + -8x2 Combine like terms: 6x + 4x = 10x 27 + 10x + -1x2 + -8x2 = -4x + 8x2 + 4x + -8x2 Combine like terms: -1x2 + -8x2 = -9x2 27 + 10x + -9x2 = -4x + 8x2 + 4x + -8x2 Reorder the terms: 27 + 10x + -9x2 = -4x + 4x + 8x2 + -8x2 Combine like terms: -4x + 4x = 0 27 + 10x + -9x2 = 0 + 8x2 + -8x2 27 + 10x + -9x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 27 + 10x + -9x2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -3 + -1.111111111x + x2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + -1.111111111x + 3 + x2 = 0 + 3 Reorder the terms: -3 + 3 + -1.111111111x + x2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -1.111111111x + x2 = 0 + 3 -1.111111111x + x2 = 0 + 3 Combine like terms: 0 + 3 = 3 -1.111111111x + x2 = 3 The x term is -1.111111111x. Take half its coefficient (-0.5555555555). Square it (0.3086419752) and add it to both sides. Add '0.3086419752' to each side of the equation. -1.111111111x + 0.3086419752 + x2 = 3 + 0.3086419752 Reorder the terms: 0.3086419752 + -1.111111111x + x2 = 3 + 0.3086419752 Combine like terms: 3 + 0.3086419752 = 3.3086419752 0.3086419752 + -1.111111111x + x2 = 3.3086419752 Factor a perfect square on the left side: (x + -0.5555555555)(x + -0.5555555555) = 3.3086419752 Calculate the square root of the right side: 1.818967283 Break this problem into two subproblems by setting (x + -0.5555555555) equal to 1.818967283 and -1.818967283.Subproblem 1
x + -0.5555555555 = 1.818967283 Simplifying x + -0.5555555555 = 1.818967283 Reorder the terms: -0.5555555555 + x = 1.818967283 Solving -0.5555555555 + x = 1.818967283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + x = 1.818967283 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + x = 1.818967283 + 0.5555555555 x = 1.818967283 + 0.5555555555 Combine like terms: 1.818967283 + 0.5555555555 = 2.3745228385 x = 2.3745228385 Simplifying x = 2.3745228385Subproblem 2
x + -0.5555555555 = -1.818967283 Simplifying x + -0.5555555555 = -1.818967283 Reorder the terms: -0.5555555555 + x = -1.818967283 Solving -0.5555555555 + x = -1.818967283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5555555555' to each side of the equation. -0.5555555555 + 0.5555555555 + x = -1.818967283 + 0.5555555555 Combine like terms: -0.5555555555 + 0.5555555555 = 0.0000000000 0.0000000000 + x = -1.818967283 + 0.5555555555 x = -1.818967283 + 0.5555555555 Combine like terms: -1.818967283 + 0.5555555555 = -1.2634117275 x = -1.2634117275 Simplifying x = -1.2634117275Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.3745228385, -1.2634117275}
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