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Simplifying 6x2 = 5x + 23 Reorder the terms: 6x2 = 23 + 5x Solving 6x2 = 23 + 5x Solving for variable 'x'. Reorder the terms: -23 + -5x + 6x2 = 23 + 5x + -23 + -5x Reorder the terms: -23 + -5x + 6x2 = 23 + -23 + 5x + -5x Combine like terms: 23 + -23 = 0 -23 + -5x + 6x2 = 0 + 5x + -5x -23 + -5x + 6x2 = 5x + -5x Combine like terms: 5x + -5x = 0 -23 + -5x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.833333333 + -0.8333333333x + x2 = 0 Move the constant term to the right: Add '3.833333333' to each side of the equation. -3.833333333 + -0.8333333333x + 3.833333333 + x2 = 0 + 3.833333333 Reorder the terms: -3.833333333 + 3.833333333 + -0.8333333333x + x2 = 0 + 3.833333333 Combine like terms: -3.833333333 + 3.833333333 = 0.000000000 0.000000000 + -0.8333333333x + x2 = 0 + 3.833333333 -0.8333333333x + x2 = 0 + 3.833333333 Combine like terms: 0 + 3.833333333 = 3.833333333 -0.8333333333x + x2 = 3.833333333 The x term is -0.8333333333x. Take half its coefficient (-0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. -0.8333333333x + 0.1736111111 + x2 = 3.833333333 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333x + x2 = 3.833333333 + 0.1736111111 Combine like terms: 3.833333333 + 0.1736111111 = 4.0069444441 0.1736111111 + -0.8333333333x + x2 = 4.0069444441 Factor a perfect square on the left side: (x + -0.4166666667)(x + -0.4166666667) = 4.0069444441 Calculate the square root of the right side: 2.001735358 Break this problem into two subproblems by setting (x + -0.4166666667) equal to 2.001735358 and -2.001735358.Subproblem 1
x + -0.4166666667 = 2.001735358 Simplifying x + -0.4166666667 = 2.001735358 Reorder the terms: -0.4166666667 + x = 2.001735358 Solving -0.4166666667 + x = 2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = 2.001735358 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = 2.001735358 + 0.4166666667 x = 2.001735358 + 0.4166666667 Combine like terms: 2.001735358 + 0.4166666667 = 2.4184020247 x = 2.4184020247 Simplifying x = 2.4184020247Subproblem 2
x + -0.4166666667 = -2.001735358 Simplifying x + -0.4166666667 = -2.001735358 Reorder the terms: -0.4166666667 + x = -2.001735358 Solving -0.4166666667 + x = -2.001735358 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + x = -2.001735358 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + x = -2.001735358 + 0.4166666667 x = -2.001735358 + 0.4166666667 Combine like terms: -2.001735358 + 0.4166666667 = -1.5850686913 x = -1.5850686913 Simplifying x = -1.5850686913Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4184020247, -1.5850686913}
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