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Simplifying 5 + 2x(3x + -4) + 6 = 2x(5x + -2) Reorder the terms: 5 + 2x(-4 + 3x) + 6 = 2x(5x + -2) 5 + (-4 * 2x + 3x * 2x) + 6 = 2x(5x + -2) 5 + (-8x + 6x2) + 6 = 2x(5x + -2) Reorder the terms: 5 + 6 + -8x + 6x2 = 2x(5x + -2) Combine like terms: 5 + 6 = 11 11 + -8x + 6x2 = 2x(5x + -2) Reorder the terms: 11 + -8x + 6x2 = 2x(-2 + 5x) 11 + -8x + 6x2 = (-2 * 2x + 5x * 2x) 11 + -8x + 6x2 = (-4x + 10x2) Solving 11 + -8x + 6x2 = -4x + 10x2 Solving for variable 'x'. Reorder the terms: 11 + -8x + 4x + 6x2 + -10x2 = -4x + 10x2 + 4x + -10x2 Combine like terms: -8x + 4x = -4x 11 + -4x + 6x2 + -10x2 = -4x + 10x2 + 4x + -10x2 Combine like terms: 6x2 + -10x2 = -4x2 11 + -4x + -4x2 = -4x + 10x2 + 4x + -10x2 Reorder the terms: 11 + -4x + -4x2 = -4x + 4x + 10x2 + -10x2 Combine like terms: -4x + 4x = 0 11 + -4x + -4x2 = 0 + 10x2 + -10x2 11 + -4x + -4x2 = 10x2 + -10x2 Combine like terms: 10x2 + -10x2 = 0 11 + -4x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -2.75 + x + x2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + x + 2.75 + x2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + x + x2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + x + x2 = 0 + 2.75 x + x2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 x + x2 = 2.75 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 2.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 2.75 + 0.25 Combine like terms: 2.75 + 0.25 = 3 1.25 + x2 = 3 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (x + 0.5) equal to 1.732050808 and -1.732050808.Subproblem 1
x + 0.5 = 1.732050808 Simplifying x + 0.5 = 1.732050808 Reorder the terms: 0.5 + x = 1.732050808 Solving 0.5 + x = 1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.732050808 + -0.5 x = 1.732050808 + -0.5 Combine like terms: 1.732050808 + -0.5 = 1.232050808 x = 1.232050808 Simplifying x = 1.232050808Subproblem 2
x + 0.5 = -1.732050808 Simplifying x + 0.5 = -1.732050808 Reorder the terms: 0.5 + x = -1.732050808 Solving 0.5 + x = -1.732050808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.732050808 + -0.5 x = -1.732050808 + -0.5 Combine like terms: -1.732050808 + -0.5 = -2.232050808 x = -2.232050808 Simplifying x = -2.232050808Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.232050808, -2.232050808}
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