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Simplifying (3x + -4)(4x + 3) = x Reorder the terms: (-4 + 3x)(4x + 3) = x Reorder the terms: (-4 + 3x)(3 + 4x) = x Multiply (-4 + 3x) * (3 + 4x) (-4(3 + 4x) + 3x * (3 + 4x)) = x ((3 * -4 + 4x * -4) + 3x * (3 + 4x)) = x ((-12 + -16x) + 3x * (3 + 4x)) = x (-12 + -16x + (3 * 3x + 4x * 3x)) = x (-12 + -16x + (9x + 12x2)) = x Combine like terms: -16x + 9x = -7x (-12 + -7x + 12x2) = x Solving -12 + -7x + 12x2 = x Solving for variable 'x'. Reorder the terms: -12 + -7x + -1x + 12x2 = x + -1x Combine like terms: -7x + -1x = -8x -12 + -8x + 12x2 = x + -1x Combine like terms: x + -1x = 0 -12 + -8x + 12x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-3 + -2x + 3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-3 + -2x + 3x2)' equal to zero and attempt to solve: Simplifying -3 + -2x + 3x2 = 0 Solving -3 + -2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + -0.6666666667x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.6666666667x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + -0.6666666667x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.6666666667x + x2 = 0 + 1 -0.6666666667x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.6666666667x + x2 = 1 The x term is -0.6666666667x. Take half its coefficient (-0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. -0.6666666667x + 0.1111111112 + x2 = 1 + 0.1111111112 Reorder the terms: 0.1111111112 + -0.6666666667x + x2 = 1 + 0.1111111112 Combine like terms: 1 + 0.1111111112 = 1.1111111112 0.1111111112 + -0.6666666667x + x2 = 1.1111111112 Factor a perfect square on the left side: (x + -0.3333333334)(x + -0.3333333334) = 1.1111111112 Calculate the square root of the right side: 1.054092553 Break this problem into two subproblems by setting (x + -0.3333333334) equal to 1.054092553 and -1.054092553.Subproblem 1
x + -0.3333333334 = 1.054092553 Simplifying x + -0.3333333334 = 1.054092553 Reorder the terms: -0.3333333334 + x = 1.054092553 Solving -0.3333333334 + x = 1.054092553 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = 1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = 1.054092553 + 0.3333333334 x = 1.054092553 + 0.3333333334 Combine like terms: 1.054092553 + 0.3333333334 = 1.3874258864 x = 1.3874258864 Simplifying x = 1.3874258864Subproblem 2
x + -0.3333333334 = -1.054092553 Simplifying x + -0.3333333334 = -1.054092553 Reorder the terms: -0.3333333334 + x = -1.054092553 Solving -0.3333333334 + x = -1.054092553 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + x = -1.054092553 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + x = -1.054092553 + 0.3333333334 x = -1.054092553 + 0.3333333334 Combine like terms: -1.054092553 + 0.3333333334 = -0.7207592196 x = -0.7207592196 Simplifying x = -0.7207592196Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3874258864, -0.7207592196}Solution
x = {1.3874258864, -0.7207592196}
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