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Simplifying (8x + -12)(3x + 28) = 180 Reorder the terms: (-12 + 8x)(3x + 28) = 180 Reorder the terms: (-12 + 8x)(28 + 3x) = 180 Multiply (-12 + 8x) * (28 + 3x) (-12(28 + 3x) + 8x * (28 + 3x)) = 180 ((28 * -12 + 3x * -12) + 8x * (28 + 3x)) = 180 ((-336 + -36x) + 8x * (28 + 3x)) = 180 (-336 + -36x + (28 * 8x + 3x * 8x)) = 180 (-336 + -36x + (224x + 24x2)) = 180 Combine like terms: -36x + 224x = 188x (-336 + 188x + 24x2) = 180 Solving -336 + 188x + 24x2 = 180 Solving for variable 'x'. Reorder the terms: -336 + -180 + 188x + 24x2 = 180 + -180 Combine like terms: -336 + -180 = -516 -516 + 188x + 24x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -516 + 188x + 24x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-129 + 47x + 6x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-129 + 47x + 6x2)' equal to zero and attempt to solve: Simplifying -129 + 47x + 6x2 = 0 Solving -129 + 47x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -21.5 + 7.833333333x + x2 = 0 Move the constant term to the right: Add '21.5' to each side of the equation. -21.5 + 7.833333333x + 21.5 + x2 = 0 + 21.5 Reorder the terms: -21.5 + 21.5 + 7.833333333x + x2 = 0 + 21.5 Combine like terms: -21.5 + 21.5 = 0.0 0.0 + 7.833333333x + x2 = 0 + 21.5 7.833333333x + x2 = 0 + 21.5 Combine like terms: 0 + 21.5 = 21.5 7.833333333x + x2 = 21.5 The x term is 7.833333333x. Take half its coefficient (3.916666667). Square it (15.34027778) and add it to both sides. Add '15.34027778' to each side of the equation. 7.833333333x + 15.34027778 + x2 = 21.5 + 15.34027778 Reorder the terms: 15.34027778 + 7.833333333x + x2 = 21.5 + 15.34027778 Combine like terms: 21.5 + 15.34027778 = 36.84027778 15.34027778 + 7.833333333x + x2 = 36.84027778 Factor a perfect square on the left side: (x + 3.916666667)(x + 3.916666667) = 36.84027778 Calculate the square root of the right side: 6.069619245 Break this problem into two subproblems by setting (x + 3.916666667) equal to 6.069619245 and -6.069619245.Subproblem 1
x + 3.916666667 = 6.069619245 Simplifying x + 3.916666667 = 6.069619245 Reorder the terms: 3.916666667 + x = 6.069619245 Solving 3.916666667 + x = 6.069619245 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.916666667' to each side of the equation. 3.916666667 + -3.916666667 + x = 6.069619245 + -3.916666667 Combine like terms: 3.916666667 + -3.916666667 = 0.000000000 0.000000000 + x = 6.069619245 + -3.916666667 x = 6.069619245 + -3.916666667 Combine like terms: 6.069619245 + -3.916666667 = 2.152952578 x = 2.152952578 Simplifying x = 2.152952578Subproblem 2
x + 3.916666667 = -6.069619245 Simplifying x + 3.916666667 = -6.069619245 Reorder the terms: 3.916666667 + x = -6.069619245 Solving 3.916666667 + x = -6.069619245 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.916666667' to each side of the equation. 3.916666667 + -3.916666667 + x = -6.069619245 + -3.916666667 Combine like terms: 3.916666667 + -3.916666667 = 0.000000000 0.000000000 + x = -6.069619245 + -3.916666667 x = -6.069619245 + -3.916666667 Combine like terms: -6.069619245 + -3.916666667 = -9.986285912 x = -9.986285912 Simplifying x = -9.986285912Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.152952578, -9.986285912}Solution
x = {2.152952578, -9.986285912}
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