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Simplifying 2x2 + 114 = 4x2 + 13x Reorder the terms: 114 + 2x2 = 4x2 + 13x Reorder the terms: 114 + 2x2 = 13x + 4x2 Solving 114 + 2x2 = 13x + 4x2 Solving for variable 'x'. Reorder the terms: 114 + -13x + 2x2 + -4x2 = 13x + 4x2 + -13x + -4x2 Combine like terms: 2x2 + -4x2 = -2x2 114 + -13x + -2x2 = 13x + 4x2 + -13x + -4x2 Reorder the terms: 114 + -13x + -2x2 = 13x + -13x + 4x2 + -4x2 Combine like terms: 13x + -13x = 0 114 + -13x + -2x2 = 0 + 4x2 + -4x2 114 + -13x + -2x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 114 + -13x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -57 + 6.5x + x2 = 0 Move the constant term to the right: Add '57' to each side of the equation. -57 + 6.5x + 57 + x2 = 0 + 57 Reorder the terms: -57 + 57 + 6.5x + x2 = 0 + 57 Combine like terms: -57 + 57 = 0 0 + 6.5x + x2 = 0 + 57 6.5x + x2 = 0 + 57 Combine like terms: 0 + 57 = 57 6.5x + x2 = 57 The x term is 6.5x. Take half its coefficient (3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. 6.5x + 10.5625 + x2 = 57 + 10.5625 Reorder the terms: 10.5625 + 6.5x + x2 = 57 + 10.5625 Combine like terms: 57 + 10.5625 = 67.5625 10.5625 + 6.5x + x2 = 67.5625 Factor a perfect square on the left side: (x + 3.25)(x + 3.25) = 67.5625 Calculate the square root of the right side: 8.219641111 Break this problem into two subproblems by setting (x + 3.25) equal to 8.219641111 and -8.219641111.Subproblem 1
x + 3.25 = 8.219641111 Simplifying x + 3.25 = 8.219641111 Reorder the terms: 3.25 + x = 8.219641111 Solving 3.25 + x = 8.219641111 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = 8.219641111 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = 8.219641111 + -3.25 x = 8.219641111 + -3.25 Combine like terms: 8.219641111 + -3.25 = 4.969641111 x = 4.969641111 Simplifying x = 4.969641111Subproblem 2
x + 3.25 = -8.219641111 Simplifying x + 3.25 = -8.219641111 Reorder the terms: 3.25 + x = -8.219641111 Solving 3.25 + x = -8.219641111 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.25' to each side of the equation. 3.25 + -3.25 + x = -8.219641111 + -3.25 Combine like terms: 3.25 + -3.25 = 0.00 0.00 + x = -8.219641111 + -3.25 x = -8.219641111 + -3.25 Combine like terms: -8.219641111 + -3.25 = -11.469641111 x = -11.469641111 Simplifying x = -11.469641111Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.969641111, -11.469641111}
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