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Simplifying (12x + 2.5)(3x + -1) = 0 Reorder the terms: (2.5 + 12x)(3x + -1) = 0 Reorder the terms: (2.5 + 12x)(-1 + 3x) = 0 Multiply (2.5 + 12x) * (-1 + 3x) (2.5(-1 + 3x) + 12x * (-1 + 3x)) = 0 ((-1 * 2.5 + 3x * 2.5) + 12x * (-1 + 3x)) = 0 ((-2.5 + 7.5x) + 12x * (-1 + 3x)) = 0 (-2.5 + 7.5x + (-1 * 12x + 3x * 12x)) = 0 (-2.5 + 7.5x + (-12x + 36x2)) = 0 Combine like terms: 7.5x + -12x = -4.5x (-2.5 + -4.5x + 36x2) = 0 Solving -2.5 + -4.5x + 36x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.06944444444 + -0.125x + x2 = 0 Move the constant term to the right: Add '0.06944444444' to each side of the equation. -0.06944444444 + -0.125x + 0.06944444444 + x2 = 0 + 0.06944444444 Reorder the terms: -0.06944444444 + 0.06944444444 + -0.125x + x2 = 0 + 0.06944444444 Combine like terms: -0.06944444444 + 0.06944444444 = 0.00000000000 0.00000000000 + -0.125x + x2 = 0 + 0.06944444444 -0.125x + x2 = 0 + 0.06944444444 Combine like terms: 0 + 0.06944444444 = 0.06944444444 -0.125x + x2 = 0.06944444444 The x term is -0.125x. Take half its coefficient (-0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. -0.125x + 0.00390625 + x2 = 0.06944444444 + 0.00390625 Reorder the terms: 0.00390625 + -0.125x + x2 = 0.06944444444 + 0.00390625 Combine like terms: 0.06944444444 + 0.00390625 = 0.07335069444 0.00390625 + -0.125x + x2 = 0.07335069444 Factor a perfect square on the left side: (x + -0.0625)(x + -0.0625) = 0.07335069444 Calculate the square root of the right side: 0.270833333 Break this problem into two subproblems by setting (x + -0.0625) equal to 0.270833333 and -0.270833333.Subproblem 1
x + -0.0625 = 0.270833333 Simplifying x + -0.0625 = 0.270833333 Reorder the terms: -0.0625 + x = 0.270833333 Solving -0.0625 + x = 0.270833333 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + x = 0.270833333 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + x = 0.270833333 + 0.0625 x = 0.270833333 + 0.0625 Combine like terms: 0.270833333 + 0.0625 = 0.333333333 x = 0.333333333 Simplifying x = 0.333333333Subproblem 2
x + -0.0625 = -0.270833333 Simplifying x + -0.0625 = -0.270833333 Reorder the terms: -0.0625 + x = -0.270833333 Solving -0.0625 + x = -0.270833333 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.0625' to each side of the equation. -0.0625 + 0.0625 + x = -0.270833333 + 0.0625 Combine like terms: -0.0625 + 0.0625 = 0.0000 0.0000 + x = -0.270833333 + 0.0625 x = -0.270833333 + 0.0625 Combine like terms: -0.270833333 + 0.0625 = -0.208333333 x = -0.208333333 Simplifying x = -0.208333333Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.333333333, -0.208333333}
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