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Simplifying (2x2 + -7x + -51) + (4x2 + 9x) = 0 Reorder the terms: (-51 + -7x + 2x2) + (4x2 + 9x) = 0 Remove parenthesis around (-51 + -7x + 2x2) -51 + -7x + 2x2 + (4x2 + 9x) = 0 Reorder the terms: -51 + -7x + 2x2 + (9x + 4x2) = 0 Remove parenthesis around (9x + 4x2) -51 + -7x + 2x2 + 9x + 4x2 = 0 Reorder the terms: -51 + -7x + 9x + 2x2 + 4x2 = 0 Combine like terms: -7x + 9x = 2x -51 + 2x + 2x2 + 4x2 = 0 Combine like terms: 2x2 + 4x2 = 6x2 -51 + 2x + 6x2 = 0 Solving -51 + 2x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -8.5 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '8.5' to each side of the equation. -8.5 + 0.3333333333x + 8.5 + x2 = 0 + 8.5 Reorder the terms: -8.5 + 8.5 + 0.3333333333x + x2 = 0 + 8.5 Combine like terms: -8.5 + 8.5 = 0.0 0.0 + 0.3333333333x + x2 = 0 + 8.5 0.3333333333x + x2 = 0 + 8.5 Combine like terms: 0 + 8.5 = 8.5 0.3333333333x + x2 = 8.5 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 = 8.5 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 8.5 + 0.02777777779 Combine like terms: 8.5 + 0.02777777779 = 8.52777777779 0.02777777779 + 0.3333333333x + x2 = 8.52777777779 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 8.52777777779 Calculate the square root of the right side: 2.920235911 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 2.920235911 and -2.920235911.Subproblem 1
x + 0.1666666667 = 2.920235911 Simplifying x + 0.1666666667 = 2.920235911 Reorder the terms: 0.1666666667 + x = 2.920235911 Solving 0.1666666667 + x = 2.920235911 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 = 2.920235911 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 2.920235911 + -0.1666666667 x = 2.920235911 + -0.1666666667 Combine like terms: 2.920235911 + -0.1666666667 = 2.7535692443 x = 2.7535692443 Simplifying x = 2.7535692443Subproblem 2
x + 0.1666666667 = -2.920235911 Simplifying x + 0.1666666667 = -2.920235911 Reorder the terms: 0.1666666667 + x = -2.920235911 Solving 0.1666666667 + x = -2.920235911 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 = -2.920235911 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -2.920235911 + -0.1666666667 x = -2.920235911 + -0.1666666667 Combine like terms: -2.920235911 + -0.1666666667 = -3.0869025777 x = -3.0869025777 Simplifying x = -3.0869025777Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.7535692443, -3.0869025777}
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