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Simplifying 6x2 + 15x + -65 = 0 Reorder the terms: -65 + 15x + 6x2 = 0 Solving -65 + 15x + 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'. -10.83333333 + 2.5x + x2 = 0 Move the constant term to the right: Add '10.83333333' to each side of the equation. -10.83333333 + 2.5x + 10.83333333 + x2 = 0 + 10.83333333 Reorder the terms: -10.83333333 + 10.83333333 + 2.5x + x2 = 0 + 10.83333333 Combine like terms: -10.83333333 + 10.83333333 = 0.00000000 0.00000000 + 2.5x + x2 = 0 + 10.83333333 2.5x + x2 = 0 + 10.83333333 Combine like terms: 0 + 10.83333333 = 10.83333333 2.5x + x2 = 10.83333333 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = 10.83333333 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = 10.83333333 + 1.5625 Combine like terms: 10.83333333 + 1.5625 = 12.39583333 1.5625 + 2.5x + x2 = 12.39583333 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 12.39583333 Calculate the square root of the right side: 3.520771695 Break this problem into two subproblems by setting (x + 1.25) equal to 3.520771695 and -3.520771695.Subproblem 1
x + 1.25 = 3.520771695 Simplifying x + 1.25 = 3.520771695 Reorder the terms: 1.25 + x = 3.520771695 Solving 1.25 + x = 3.520771695 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 3.520771695 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 3.520771695 + -1.25 x = 3.520771695 + -1.25 Combine like terms: 3.520771695 + -1.25 = 2.270771695 x = 2.270771695 Simplifying x = 2.270771695Subproblem 2
x + 1.25 = -3.520771695 Simplifying x + 1.25 = -3.520771695 Reorder the terms: 1.25 + x = -3.520771695 Solving 1.25 + x = -3.520771695 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -3.520771695 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -3.520771695 + -1.25 x = -3.520771695 + -1.25 Combine like terms: -3.520771695 + -1.25 = -4.770771695 x = -4.770771695 Simplifying x = -4.770771695Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.270771695, -4.770771695}
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