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Simplifying 0 = 4x2 + 22x + -54 Reorder the terms: 0 = -54 + 22x + 4x2 Solving 0 = -54 + 22x + 4x2 Solving for variable 'x'. Combine like terms: 0 + 54 = 54 54 + -22x + -4x2 = -54 + 22x + 4x2 + 54 + -22x + -4x2 Reorder the terms: 54 + -22x + -4x2 = -54 + 54 + 22x + -22x + 4x2 + -4x2 Combine like terms: -54 + 54 = 0 54 + -22x + -4x2 = 0 + 22x + -22x + 4x2 + -4x2 54 + -22x + -4x2 = 22x + -22x + 4x2 + -4x2 Combine like terms: 22x + -22x = 0 54 + -22x + -4x2 = 0 + 4x2 + -4x2 54 + -22x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 54 + -22x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(27 + -11x + -2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(27 + -11x + -2x2)' equal to zero and attempt to solve: Simplifying 27 + -11x + -2x2 = 0 Solving 27 + -11x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -13.5 + 5.5x + x2 = 0 Move the constant term to the right: Add '13.5' to each side of the equation. -13.5 + 5.5x + 13.5 + x2 = 0 + 13.5 Reorder the terms: -13.5 + 13.5 + 5.5x + x2 = 0 + 13.5 Combine like terms: -13.5 + 13.5 = 0.0 0.0 + 5.5x + x2 = 0 + 13.5 5.5x + x2 = 0 + 13.5 Combine like terms: 0 + 13.5 = 13.5 5.5x + x2 = 13.5 The x term is 5.5x. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5x + 7.5625 + x2 = 13.5 + 7.5625 Reorder the terms: 7.5625 + 5.5x + x2 = 13.5 + 7.5625 Combine like terms: 13.5 + 7.5625 = 21.0625 7.5625 + 5.5x + x2 = 21.0625 Factor a perfect square on the left side: (x + 2.75)(x + 2.75) = 21.0625 Calculate the square root of the right side: 4.589389938 Break this problem into two subproblems by setting (x + 2.75) equal to 4.589389938 and -4.589389938.Subproblem 1
x + 2.75 = 4.589389938 Simplifying x + 2.75 = 4.589389938 Reorder the terms: 2.75 + x = 4.589389938 Solving 2.75 + x = 4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = 4.589389938 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = 4.589389938 + -2.75 x = 4.589389938 + -2.75 Combine like terms: 4.589389938 + -2.75 = 1.839389938 x = 1.839389938 Simplifying x = 1.839389938Subproblem 2
x + 2.75 = -4.589389938 Simplifying x + 2.75 = -4.589389938 Reorder the terms: 2.75 + x = -4.589389938 Solving 2.75 + x = -4.589389938 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + x = -4.589389938 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + x = -4.589389938 + -2.75 x = -4.589389938 + -2.75 Combine like terms: -4.589389938 + -2.75 = -7.339389938 x = -7.339389938 Simplifying x = -7.339389938Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.839389938, -7.339389938}Solution
x = {1.839389938, -7.339389938}
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