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Simplifying 37 + x(3x + 51) = 180 Reorder the terms: 37 + x(51 + 3x) = 180 37 + (51 * x + 3x * x) = 180 37 + (51x + 3x2) = 180 Solving 37 + 51x + 3x2 = 180 Solving for variable 'x'. Reorder the terms: 37 + -180 + 51x + 3x2 = 180 + -180 Combine like terms: 37 + -180 = -143 -143 + 51x + 3x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -143 + 51x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -47.66666667 + 17x + x2 = 0 Move the constant term to the right: Add '47.66666667' to each side of the equation. -47.66666667 + 17x + 47.66666667 + x2 = 0 + 47.66666667 Reorder the terms: -47.66666667 + 47.66666667 + 17x + x2 = 0 + 47.66666667 Combine like terms: -47.66666667 + 47.66666667 = 0.00000000 0.00000000 + 17x + x2 = 0 + 47.66666667 17x + x2 = 0 + 47.66666667 Combine like terms: 0 + 47.66666667 = 47.66666667 17x + x2 = 47.66666667 The x term is 17x. Take half its coefficient (8.5). Square it (72.25) and add it to both sides. Add '72.25' to each side of the equation. 17x + 72.25 + x2 = 47.66666667 + 72.25 Reorder the terms: 72.25 + 17x + x2 = 47.66666667 + 72.25 Combine like terms: 47.66666667 + 72.25 = 119.91666667 72.25 + 17x + x2 = 119.91666667 Factor a perfect square on the left side: (x + 8.5)(x + 8.5) = 119.91666667 Calculate the square root of the right side: 10.950646861 Break this problem into two subproblems by setting (x + 8.5) equal to 10.950646861 and -10.950646861.Subproblem 1
x + 8.5 = 10.950646861 Simplifying x + 8.5 = 10.950646861 Reorder the terms: 8.5 + x = 10.950646861 Solving 8.5 + x = 10.950646861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = 10.950646861 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = 10.950646861 + -8.5 x = 10.950646861 + -8.5 Combine like terms: 10.950646861 + -8.5 = 2.450646861 x = 2.450646861 Simplifying x = 2.450646861Subproblem 2
x + 8.5 = -10.950646861 Simplifying x + 8.5 = -10.950646861 Reorder the terms: 8.5 + x = -10.950646861 Solving 8.5 + x = -10.950646861 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8.5' to each side of the equation. 8.5 + -8.5 + x = -10.950646861 + -8.5 Combine like terms: 8.5 + -8.5 = 0.0 0.0 + x = -10.950646861 + -8.5 x = -10.950646861 + -8.5 Combine like terms: -10.950646861 + -8.5 = -19.450646861 x = -19.450646861 Simplifying x = -19.450646861Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.450646861, -19.450646861}
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