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Simplifying 7x2 + 3x + 91 = 180 Reorder the terms: 91 + 3x + 7x2 = 180 Solving 91 + 3x + 7x2 = 180 Solving for variable 'x'. Reorder the terms: 91 + -180 + 3x + 7x2 = 180 + -180 Combine like terms: 91 + -180 = -89 -89 + 3x + 7x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -89 + 3x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -12.71428571 + 0.4285714286x + x2 = 0 Move the constant term to the right: Add '12.71428571' to each side of the equation. -12.71428571 + 0.4285714286x + 12.71428571 + x2 = 0 + 12.71428571 Reorder the terms: -12.71428571 + 12.71428571 + 0.4285714286x + x2 = 0 + 12.71428571 Combine like terms: -12.71428571 + 12.71428571 = 0.00000000 0.00000000 + 0.4285714286x + x2 = 0 + 12.71428571 0.4285714286x + x2 = 0 + 12.71428571 Combine like terms: 0 + 12.71428571 = 12.71428571 0.4285714286x + x2 = 12.71428571 The x term is 0.4285714286x. Take half its coefficient (0.2142857143). Square it (0.04591836735) and add it to both sides. Add '0.04591836735' to each side of the equation. 0.4285714286x + 0.04591836735 + x2 = 12.71428571 + 0.04591836735 Reorder the terms: 0.04591836735 + 0.4285714286x + x2 = 12.71428571 + 0.04591836735 Combine like terms: 12.71428571 + 0.04591836735 = 12.76020407735 0.04591836735 + 0.4285714286x + x2 = 12.76020407735 Factor a perfect square on the left side: (x + 0.2142857143)(x + 0.2142857143) = 12.76020407735 Calculate the square root of the right side: 3.572142785 Break this problem into two subproblems by setting (x + 0.2142857143) equal to 3.572142785 and -3.572142785.Subproblem 1
x + 0.2142857143 = 3.572142785 Simplifying x + 0.2142857143 = 3.572142785 Reorder the terms: 0.2142857143 + x = 3.572142785 Solving 0.2142857143 + x = 3.572142785 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + x = 3.572142785 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + x = 3.572142785 + -0.2142857143 x = 3.572142785 + -0.2142857143 Combine like terms: 3.572142785 + -0.2142857143 = 3.3578570707 x = 3.3578570707 Simplifying x = 3.3578570707Subproblem 2
x + 0.2142857143 = -3.572142785 Simplifying x + 0.2142857143 = -3.572142785 Reorder the terms: 0.2142857143 + x = -3.572142785 Solving 0.2142857143 + x = -3.572142785 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.2142857143' to each side of the equation. 0.2142857143 + -0.2142857143 + x = -3.572142785 + -0.2142857143 Combine like terms: 0.2142857143 + -0.2142857143 = 0.0000000000 0.0000000000 + x = -3.572142785 + -0.2142857143 x = -3.572142785 + -0.2142857143 Combine like terms: -3.572142785 + -0.2142857143 = -3.7864284993 x = -3.7864284993 Simplifying x = -3.7864284993Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.3578570707, -3.7864284993}
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