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Simplifying 180 = (6x + -27)(4x + 7) Reorder the terms: 180 = (-27 + 6x)(4x + 7) Reorder the terms: 180 = (-27 + 6x)(7 + 4x) Multiply (-27 + 6x) * (7 + 4x) 180 = (-27(7 + 4x) + 6x * (7 + 4x)) 180 = ((7 * -27 + 4x * -27) + 6x * (7 + 4x)) 180 = ((-189 + -108x) + 6x * (7 + 4x)) 180 = (-189 + -108x + (7 * 6x + 4x * 6x)) 180 = (-189 + -108x + (42x + 24x2)) Combine like terms: -108x + 42x = -66x 180 = (-189 + -66x + 24x2) Solving 180 = -189 + -66x + 24x2 Solving for variable 'x'. Combine like terms: 180 + 189 = 369 369 + 66x + -24x2 = -189 + -66x + 24x2 + 189 + 66x + -24x2 Reorder the terms: 369 + 66x + -24x2 = -189 + 189 + -66x + 66x + 24x2 + -24x2 Combine like terms: -189 + 189 = 0 369 + 66x + -24x2 = 0 + -66x + 66x + 24x2 + -24x2 369 + 66x + -24x2 = -66x + 66x + 24x2 + -24x2 Combine like terms: -66x + 66x = 0 369 + 66x + -24x2 = 0 + 24x2 + -24x2 369 + 66x + -24x2 = 24x2 + -24x2 Combine like terms: 24x2 + -24x2 = 0 369 + 66x + -24x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(123 + 22x + -8x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(123 + 22x + -8x2)' equal to zero and attempt to solve: Simplifying 123 + 22x + -8x2 = 0 Solving 123 + 22x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -15.375 + -2.75x + x2 = 0 Move the constant term to the right: Add '15.375' to each side of the equation. -15.375 + -2.75x + 15.375 + x2 = 0 + 15.375 Reorder the terms: -15.375 + 15.375 + -2.75x + x2 = 0 + 15.375 Combine like terms: -15.375 + 15.375 = 0.000 0.000 + -2.75x + x2 = 0 + 15.375 -2.75x + x2 = 0 + 15.375 Combine like terms: 0 + 15.375 = 15.375 -2.75x + x2 = 15.375 The x term is -2.75x. Take half its coefficient (-1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. -2.75x + 1.890625 + x2 = 15.375 + 1.890625 Reorder the terms: 1.890625 + -2.75x + x2 = 15.375 + 1.890625 Combine like terms: 15.375 + 1.890625 = 17.265625 1.890625 + -2.75x + x2 = 17.265625 Factor a perfect square on the left side: (x + -1.375)(x + -1.375) = 17.265625 Calculate the square root of the right side: 4.155192535 Break this problem into two subproblems by setting (x + -1.375) equal to 4.155192535 and -4.155192535.Subproblem 1
x + -1.375 = 4.155192535 Simplifying x + -1.375 = 4.155192535 Reorder the terms: -1.375 + x = 4.155192535 Solving -1.375 + x = 4.155192535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = 4.155192535 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = 4.155192535 + 1.375 x = 4.155192535 + 1.375 Combine like terms: 4.155192535 + 1.375 = 5.530192535 x = 5.530192535 Simplifying x = 5.530192535Subproblem 2
x + -1.375 = -4.155192535 Simplifying x + -1.375 = -4.155192535 Reorder the terms: -1.375 + x = -4.155192535 Solving -1.375 + x = -4.155192535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.375' to each side of the equation. -1.375 + 1.375 + x = -4.155192535 + 1.375 Combine like terms: -1.375 + 1.375 = 0.000 0.000 + x = -4.155192535 + 1.375 x = -4.155192535 + 1.375 Combine like terms: -4.155192535 + 1.375 = -2.780192535 x = -2.780192535 Simplifying x = -2.780192535Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.530192535, -2.780192535}Solution
x = {5.530192535, -2.780192535}
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