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Simplifying 12x2 + -76x + 78 = 0 Reorder the terms: 78 + -76x + 12x2 = 0 Solving 78 + -76x + 12x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(39 + -38x + 6x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(39 + -38x + 6x2)' equal to zero and attempt to solve: Simplifying 39 + -38x + 6x2 = 0 Solving 39 + -38x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 6.5 + -6.333333333x + x2 = 0 Move the constant term to the right: Add '-6.5' to each side of the equation. 6.5 + -6.333333333x + -6.5 + x2 = 0 + -6.5 Reorder the terms: 6.5 + -6.5 + -6.333333333x + x2 = 0 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + -6.333333333x + x2 = 0 + -6.5 -6.333333333x + x2 = 0 + -6.5 Combine like terms: 0 + -6.5 = -6.5 -6.333333333x + x2 = -6.5 The x term is -6.333333333x. Take half its coefficient (-3.166666667). Square it (10.02777778) and add it to both sides. Add '10.02777778' to each side of the equation. -6.333333333x + 10.02777778 + x2 = -6.5 + 10.02777778 Reorder the terms: 10.02777778 + -6.333333333x + x2 = -6.5 + 10.02777778 Combine like terms: -6.5 + 10.02777778 = 3.52777778 10.02777778 + -6.333333333x + x2 = 3.52777778 Factor a perfect square on the left side: (x + -3.166666667)(x + -3.166666667) = 3.52777778 Calculate the square root of the right side: 1.878237946 Break this problem into two subproblems by setting (x + -3.166666667) equal to 1.878237946 and -1.878237946.Subproblem 1
x + -3.166666667 = 1.878237946 Simplifying x + -3.166666667 = 1.878237946 Reorder the terms: -3.166666667 + x = 1.878237946 Solving -3.166666667 + x = 1.878237946 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = 1.878237946 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = 1.878237946 + 3.166666667 x = 1.878237946 + 3.166666667 Combine like terms: 1.878237946 + 3.166666667 = 5.044904613 x = 5.044904613 Simplifying x = 5.044904613Subproblem 2
x + -3.166666667 = -1.878237946 Simplifying x + -3.166666667 = -1.878237946 Reorder the terms: -3.166666667 + x = -1.878237946 Solving -3.166666667 + x = -1.878237946 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.166666667' to each side of the equation. -3.166666667 + 3.166666667 + x = -1.878237946 + 3.166666667 Combine like terms: -3.166666667 + 3.166666667 = 0.000000000 0.000000000 + x = -1.878237946 + 3.166666667 x = -1.878237946 + 3.166666667 Combine like terms: -1.878237946 + 3.166666667 = 1.288428721 x = 1.288428721 Simplifying x = 1.288428721Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.044904613, 1.288428721}Solution
x = {5.044904613, 1.288428721}
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