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Simplifying 97000 = 2437x2 + 5713x + 5900 Reorder the terms: 97000 = 5900 + 5713x + 2437x2 Solving 97000 = 5900 + 5713x + 2437x2 Solving for variable 'x'. Combine like terms: 97000 + -5900 = 91100 91100 + -5713x + -2437x2 = 5900 + 5713x + 2437x2 + -5900 + -5713x + -2437x2 Reorder the terms: 91100 + -5713x + -2437x2 = 5900 + -5900 + 5713x + -5713x + 2437x2 + -2437x2 Combine like terms: 5900 + -5900 = 0 91100 + -5713x + -2437x2 = 0 + 5713x + -5713x + 2437x2 + -2437x2 91100 + -5713x + -2437x2 = 5713x + -5713x + 2437x2 + -2437x2 Combine like terms: 5713x + -5713x = 0 91100 + -5713x + -2437x2 = 0 + 2437x2 + -2437x2 91100 + -5713x + -2437x2 = 2437x2 + -2437x2 Combine like terms: 2437x2 + -2437x2 = 0 91100 + -5713x + -2437x2 = 0 Begin completing the square. Divide all terms by -2437 the coefficient of the squared term: Divide each side by '-2437'. -37.38202708 + 2.344275749x + x2 = 0 Move the constant term to the right: Add '37.38202708' to each side of the equation. -37.38202708 + 2.344275749x + 37.38202708 + x2 = 0 + 37.38202708 Reorder the terms: -37.38202708 + 37.38202708 + 2.344275749x + x2 = 0 + 37.38202708 Combine like terms: -37.38202708 + 37.38202708 = 0.00000000 0.00000000 + 2.344275749x + x2 = 0 + 37.38202708 2.344275749x + x2 = 0 + 37.38202708 Combine like terms: 0 + 37.38202708 = 37.38202708 2.344275749x + x2 = 37.38202708 The x term is 2.344275749x. Take half its coefficient (1.172137875). Square it (1.373907198) and add it to both sides. Add '1.373907198' to each side of the equation. 2.344275749x + 1.373907198 + x2 = 37.38202708 + 1.373907198 Reorder the terms: 1.373907198 + 2.344275749x + x2 = 37.38202708 + 1.373907198 Combine like terms: 37.38202708 + 1.373907198 = 38.755934278 1.373907198 + 2.344275749x + x2 = 38.755934278 Factor a perfect square on the left side: (x + 1.172137875)(x + 1.172137875) = 38.755934278 Calculate the square root of the right side: 6.225426433 Break this problem into two subproblems by setting (x + 1.172137875) equal to 6.225426433 and -6.225426433.Subproblem 1
x + 1.172137875 = 6.225426433 Simplifying x + 1.172137875 = 6.225426433 Reorder the terms: 1.172137875 + x = 6.225426433 Solving 1.172137875 + x = 6.225426433 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.172137875' to each side of the equation. 1.172137875 + -1.172137875 + x = 6.225426433 + -1.172137875 Combine like terms: 1.172137875 + -1.172137875 = 0.000000000 0.000000000 + x = 6.225426433 + -1.172137875 x = 6.225426433 + -1.172137875 Combine like terms: 6.225426433 + -1.172137875 = 5.053288558 x = 5.053288558 Simplifying x = 5.053288558Subproblem 2
x + 1.172137875 = -6.225426433 Simplifying x + 1.172137875 = -6.225426433 Reorder the terms: 1.172137875 + x = -6.225426433 Solving 1.172137875 + x = -6.225426433 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.172137875' to each side of the equation. 1.172137875 + -1.172137875 + x = -6.225426433 + -1.172137875 Combine like terms: 1.172137875 + -1.172137875 = 0.000000000 0.000000000 + x = -6.225426433 + -1.172137875 x = -6.225426433 + -1.172137875 Combine like terms: -6.225426433 + -1.172137875 = -7.397564308 x = -7.397564308 Simplifying x = -7.397564308Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.053288558, -7.397564308}
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