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Simplifying (2x2 + -4x + -16) + (7x2 + 7x) = 0 Reorder the terms: (-16 + -4x + 2x2) + (7x2 + 7x) = 0 Remove parenthesis around (-16 + -4x + 2x2) -16 + -4x + 2x2 + (7x2 + 7x) = 0 Reorder the terms: -16 + -4x + 2x2 + (7x + 7x2) = 0 Remove parenthesis around (7x + 7x2) -16 + -4x + 2x2 + 7x + 7x2 = 0 Reorder the terms: -16 + -4x + 7x + 2x2 + 7x2 = 0 Combine like terms: -4x + 7x = 3x -16 + 3x + 2x2 + 7x2 = 0 Combine like terms: 2x2 + 7x2 = 9x2 -16 + 3x + 9x2 = 0 Solving -16 + 3x + 9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.777777778 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '1.777777778' to each side of the equation. -1.777777778 + 0.3333333333x + 1.777777778 + x2 = 0 + 1.777777778 Reorder the terms: -1.777777778 + 1.777777778 + 0.3333333333x + x2 = 0 + 1.777777778 Combine like terms: -1.777777778 + 1.777777778 = 0.000000000 0.000000000 + 0.3333333333x + x2 = 0 + 1.777777778 0.3333333333x + x2 = 0 + 1.777777778 Combine like terms: 0 + 1.777777778 = 1.777777778 0.3333333333x + x2 = 1.777777778 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 1.777777778 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 1.777777778 + 0.02777777779 Combine like terms: 1.777777778 + 0.02777777779 = 1.80555555579 0.02777777779 + 0.3333333333x + x2 = 1.80555555579 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 1.80555555579 Calculate the square root of the right side: 1.343709625 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 1.343709625 and -1.343709625.Subproblem 1
x + 0.1666666667 = 1.343709625 Simplifying x + 0.1666666667 = 1.343709625 Reorder the terms: 0.1666666667 + x = 1.343709625 Solving 0.1666666667 + x = 1.343709625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 1.343709625 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 1.343709625 + -0.1666666667 x = 1.343709625 + -0.1666666667 Combine like terms: 1.343709625 + -0.1666666667 = 1.1770429583 x = 1.1770429583 Simplifying x = 1.1770429583Subproblem 2
x + 0.1666666667 = -1.343709625 Simplifying x + 0.1666666667 = -1.343709625 Reorder the terms: 0.1666666667 + x = -1.343709625 Solving 0.1666666667 + x = -1.343709625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -1.343709625 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -1.343709625 + -0.1666666667 x = -1.343709625 + -0.1666666667 Combine like terms: -1.343709625 + -0.1666666667 = -1.5103762917 x = -1.5103762917 Simplifying x = -1.5103762917Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.1770429583, -1.5103762917}
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