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Simplifying 0 = 12x2 + -172x + 420 Reorder the terms: 0 = 420 + -172x + 12x2 Solving 0 = 420 + -172x + 12x2 Solving for variable 'x'. Combine like terms: 0 + -420 = -420 -420 + 172x + -12x2 = 420 + -172x + 12x2 + -420 + 172x + -12x2 Reorder the terms: -420 + 172x + -12x2 = 420 + -420 + -172x + 172x + 12x2 + -12x2 Combine like terms: 420 + -420 = 0 -420 + 172x + -12x2 = 0 + -172x + 172x + 12x2 + -12x2 -420 + 172x + -12x2 = -172x + 172x + 12x2 + -12x2 Combine like terms: -172x + 172x = 0 -420 + 172x + -12x2 = 0 + 12x2 + -12x2 -420 + 172x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 -420 + 172x + -12x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-105 + 43x + -3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-105 + 43x + -3x2)' equal to zero and attempt to solve: Simplifying -105 + 43x + -3x2 = 0 Solving -105 + 43x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 35 + -14.33333333x + x2 = 0 Move the constant term to the right: Add '-35' to each side of the equation. 35 + -14.33333333x + -35 + x2 = 0 + -35 Reorder the terms: 35 + -35 + -14.33333333x + x2 = 0 + -35 Combine like terms: 35 + -35 = 0 0 + -14.33333333x + x2 = 0 + -35 -14.33333333x + x2 = 0 + -35 Combine like terms: 0 + -35 = -35 -14.33333333x + x2 = -35 The x term is -14.33333333x. Take half its coefficient (-7.166666665). Square it (51.36111109) and add it to both sides. Add '51.36111109' to each side of the equation. -14.33333333x + 51.36111109 + x2 = -35 + 51.36111109 Reorder the terms: 51.36111109 + -14.33333333x + x2 = -35 + 51.36111109 Combine like terms: -35 + 51.36111109 = 16.36111109 51.36111109 + -14.33333333x + x2 = 16.36111109 Factor a perfect square on the left side: (x + -7.166666665)(x + -7.166666665) = 16.36111109 Calculate the square root of the right side: 4.044887031 Break this problem into two subproblems by setting (x + -7.166666665) equal to 4.044887031 and -4.044887031.Subproblem 1
x + -7.166666665 = 4.044887031 Simplifying x + -7.166666665 = 4.044887031 Reorder the terms: -7.166666665 + x = 4.044887031 Solving -7.166666665 + x = 4.044887031 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.166666665' to each side of the equation. -7.166666665 + 7.166666665 + x = 4.044887031 + 7.166666665 Combine like terms: -7.166666665 + 7.166666665 = 0.000000000 0.000000000 + x = 4.044887031 + 7.166666665 x = 4.044887031 + 7.166666665 Combine like terms: 4.044887031 + 7.166666665 = 11.211553696 x = 11.211553696 Simplifying x = 11.211553696Subproblem 2
x + -7.166666665 = -4.044887031 Simplifying x + -7.166666665 = -4.044887031 Reorder the terms: -7.166666665 + x = -4.044887031 Solving -7.166666665 + x = -4.044887031 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '7.166666665' to each side of the equation. -7.166666665 + 7.166666665 + x = -4.044887031 + 7.166666665 Combine like terms: -7.166666665 + 7.166666665 = 0.000000000 0.000000000 + x = -4.044887031 + 7.166666665 x = -4.044887031 + 7.166666665 Combine like terms: -4.044887031 + 7.166666665 = 3.121779634 x = 3.121779634 Simplifying x = 3.121779634Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.211553696, 3.121779634}Solution
x = {11.211553696, 3.121779634}
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