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Simplifying 6x + 1 = (4x + 5)(2x + -4) Reorder the terms: 1 + 6x = (4x + 5)(2x + -4) Reorder the terms: 1 + 6x = (5 + 4x)(2x + -4) Reorder the terms: 1 + 6x = (5 + 4x)(-4 + 2x) Multiply (5 + 4x) * (-4 + 2x) 1 + 6x = (5(-4 + 2x) + 4x * (-4 + 2x)) 1 + 6x = ((-4 * 5 + 2x * 5) + 4x * (-4 + 2x)) 1 + 6x = ((-20 + 10x) + 4x * (-4 + 2x)) 1 + 6x = (-20 + 10x + (-4 * 4x + 2x * 4x)) 1 + 6x = (-20 + 10x + (-16x + 8x2)) Combine like terms: 10x + -16x = -6x 1 + 6x = (-20 + -6x + 8x2) Solving 1 + 6x = -20 + -6x + 8x2 Solving for variable 'x'. Reorder the terms: 1 + 20 + 6x + 6x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2 Combine like terms: 1 + 20 = 21 21 + 6x + 6x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2 Combine like terms: 6x + 6x = 12x 21 + 12x + -8x2 = -20 + -6x + 8x2 + 20 + 6x + -8x2 Reorder the terms: 21 + 12x + -8x2 = -20 + 20 + -6x + 6x + 8x2 + -8x2 Combine like terms: -20 + 20 = 0 21 + 12x + -8x2 = 0 + -6x + 6x + 8x2 + -8x2 21 + 12x + -8x2 = -6x + 6x + 8x2 + -8x2 Combine like terms: -6x + 6x = 0 21 + 12x + -8x2 = 0 + 8x2 + -8x2 21 + 12x + -8x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 21 + 12x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -2.625 + -1.5x + x2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + -1.5x + 2.625 + x2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + -1.5x + x2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + -1.5x + x2 = 0 + 2.625 -1.5x + x2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 -1.5x + x2 = 2.625 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 2.625 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 2.625 + 0.5625 Combine like terms: 2.625 + 0.5625 = 3.1875 0.5625 + -1.5x + x2 = 3.1875 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 3.1875 Calculate the square root of the right side: 1.785357107 Break this problem into two subproblems by setting (x + -0.75) equal to 1.785357107 and -1.785357107.Subproblem 1
x + -0.75 = 1.785357107 Simplifying x + -0.75 = 1.785357107 Reorder the terms: -0.75 + x = 1.785357107 Solving -0.75 + x = 1.785357107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.785357107 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.785357107 + 0.75 x = 1.785357107 + 0.75 Combine like terms: 1.785357107 + 0.75 = 2.535357107 x = 2.535357107 Simplifying x = 2.535357107Subproblem 2
x + -0.75 = -1.785357107 Simplifying x + -0.75 = -1.785357107 Reorder the terms: -0.75 + x = -1.785357107 Solving -0.75 + x = -1.785357107 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.785357107 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.785357107 + 0.75 x = -1.785357107 + 0.75 Combine like terms: -1.785357107 + 0.75 = -1.035357107 x = -1.035357107 Simplifying x = -1.035357107Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.535357107, -1.035357107}
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