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Simplifying (2x + -3)(x + -4) = 21(x + -2) Reorder the terms: (-3 + 2x)(x + -4) = 21(x + -2) Reorder the terms: (-3 + 2x)(-4 + x) = 21(x + -2) Multiply (-3 + 2x) * (-4 + x) (-3(-4 + x) + 2x * (-4 + x)) = 21(x + -2) ((-4 * -3 + x * -3) + 2x * (-4 + x)) = 21(x + -2) ((12 + -3x) + 2x * (-4 + x)) = 21(x + -2) (12 + -3x + (-4 * 2x + x * 2x)) = 21(x + -2) (12 + -3x + (-8x + 2x2)) = 21(x + -2) Combine like terms: -3x + -8x = -11x (12 + -11x + 2x2) = 21(x + -2) Reorder the terms: 12 + -11x + 2x2 = 21(-2 + x) 12 + -11x + 2x2 = (-2 * 21 + x * 21) 12 + -11x + 2x2 = (-42 + 21x) Solving 12 + -11x + 2x2 = -42 + 21x Solving for variable 'x'. Reorder the terms: 12 + 42 + -11x + -21x + 2x2 = -42 + 21x + 42 + -21x Combine like terms: 12 + 42 = 54 54 + -11x + -21x + 2x2 = -42 + 21x + 42 + -21x Combine like terms: -11x + -21x = -32x 54 + -32x + 2x2 = -42 + 21x + 42 + -21x Reorder the terms: 54 + -32x + 2x2 = -42 + 42 + 21x + -21x Combine like terms: -42 + 42 = 0 54 + -32x + 2x2 = 0 + 21x + -21x 54 + -32x + 2x2 = 21x + -21x Combine like terms: 21x + -21x = 0 54 + -32x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(27 + -16x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(27 + -16x + x2)' equal to zero and attempt to solve: Simplifying 27 + -16x + x2 = 0 Solving 27 + -16x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-27' to each side of the equation. 27 + -16x + -27 + x2 = 0 + -27 Reorder the terms: 27 + -27 + -16x + x2 = 0 + -27 Combine like terms: 27 + -27 = 0 0 + -16x + x2 = 0 + -27 -16x + x2 = 0 + -27 Combine like terms: 0 + -27 = -27 -16x + x2 = -27 The x term is -16x. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16x + 64 + x2 = -27 + 64 Reorder the terms: 64 + -16x + x2 = -27 + 64 Combine like terms: -27 + 64 = 37 64 + -16x + x2 = 37 Factor a perfect square on the left side: (x + -8)(x + -8) = 37 Calculate the square root of the right side: 6.08276253 Break this problem into two subproblems by setting (x + -8) equal to 6.08276253 and -6.08276253.Subproblem 1
x + -8 = 6.08276253 Simplifying x + -8 = 6.08276253 Reorder the terms: -8 + x = 6.08276253 Solving -8 + x = 6.08276253 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = 6.08276253 + 8 Combine like terms: -8 + 8 = 0 0 + x = 6.08276253 + 8 x = 6.08276253 + 8 Combine like terms: 6.08276253 + 8 = 14.08276253 x = 14.08276253 Simplifying x = 14.08276253Subproblem 2
x + -8 = -6.08276253 Simplifying x + -8 = -6.08276253 Reorder the terms: -8 + x = -6.08276253 Solving -8 + x = -6.08276253 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + x = -6.08276253 + 8 Combine like terms: -8 + 8 = 0 0 + x = -6.08276253 + 8 x = -6.08276253 + 8 Combine like terms: -6.08276253 + 8 = 1.91723747 x = 1.91723747 Simplifying x = 1.91723747Solution
The solution to the problem is based on the solutions from the subproblems. x = {14.08276253, 1.91723747}Solution
x = {14.08276253, 1.91723747}
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