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Simplifying -56x2 + -312x + 180 = 0 Reorder the terms: 180 + -312x + -56x2 = 0 Solving 180 + -312x + -56x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(45 + -78x + -14x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(45 + -78x + -14x2)' equal to zero and attempt to solve: Simplifying 45 + -78x + -14x2 = 0 Solving 45 + -78x + -14x2 = 0 Begin completing the square. Divide all terms by -14 the coefficient of the squared term: Divide each side by '-14'. -3.214285714 + 5.571428571x + x2 = 0 Move the constant term to the right: Add '3.214285714' to each side of the equation. -3.214285714 + 5.571428571x + 3.214285714 + x2 = 0 + 3.214285714 Reorder the terms: -3.214285714 + 3.214285714 + 5.571428571x + x2 = 0 + 3.214285714 Combine like terms: -3.214285714 + 3.214285714 = 0.000000000 0.000000000 + 5.571428571x + x2 = 0 + 3.214285714 5.571428571x + x2 = 0 + 3.214285714 Combine like terms: 0 + 3.214285714 = 3.214285714 5.571428571x + x2 = 3.214285714 The x term is 5.571428571x. Take half its coefficient (2.785714286). Square it (7.760204083) and add it to both sides. Add '7.760204083' to each side of the equation. 5.571428571x + 7.760204083 + x2 = 3.214285714 + 7.760204083 Reorder the terms: 7.760204083 + 5.571428571x + x2 = 3.214285714 + 7.760204083 Combine like terms: 3.214285714 + 7.760204083 = 10.974489797 7.760204083 + 5.571428571x + x2 = 10.974489797 Factor a perfect square on the left side: (x + 2.785714286)(x + 2.785714286) = 10.974489797 Calculate the square root of the right side: 3.31277675 Break this problem into two subproblems by setting (x + 2.785714286) equal to 3.31277675 and -3.31277675.Subproblem 1
x + 2.785714286 = 3.31277675 Simplifying x + 2.785714286 = 3.31277675 Reorder the terms: 2.785714286 + x = 3.31277675 Solving 2.785714286 + x = 3.31277675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.785714286' to each side of the equation. 2.785714286 + -2.785714286 + x = 3.31277675 + -2.785714286 Combine like terms: 2.785714286 + -2.785714286 = 0.000000000 0.000000000 + x = 3.31277675 + -2.785714286 x = 3.31277675 + -2.785714286 Combine like terms: 3.31277675 + -2.785714286 = 0.527062464 x = 0.527062464 Simplifying x = 0.527062464Subproblem 2
x + 2.785714286 = -3.31277675 Simplifying x + 2.785714286 = -3.31277675 Reorder the terms: 2.785714286 + x = -3.31277675 Solving 2.785714286 + x = -3.31277675 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.785714286' to each side of the equation. 2.785714286 + -2.785714286 + x = -3.31277675 + -2.785714286 Combine like terms: 2.785714286 + -2.785714286 = 0.000000000 0.000000000 + x = -3.31277675 + -2.785714286 x = -3.31277675 + -2.785714286 Combine like terms: -3.31277675 + -2.785714286 = -6.098491036 x = -6.098491036 Simplifying x = -6.098491036Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.527062464, -6.098491036}Solution
x = {0.527062464, -6.098491036}
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