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Simplifying 14x2 + 12x = 65 Reorder the terms: 12x + 14x2 = 65 Solving 12x + 14x2 = 65 Solving for variable 'x'. Reorder the terms: -65 + 12x + 14x2 = 65 + -65 Combine like terms: 65 + -65 = 0 -65 + 12x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -4.642857143 + 0.8571428571x + x2 = 0 Move the constant term to the right: Add '4.642857143' to each side of the equation. -4.642857143 + 0.8571428571x + 4.642857143 + x2 = 0 + 4.642857143 Reorder the terms: -4.642857143 + 4.642857143 + 0.8571428571x + x2 = 0 + 4.642857143 Combine like terms: -4.642857143 + 4.642857143 = 0.000000000 0.000000000 + 0.8571428571x + x2 = 0 + 4.642857143 0.8571428571x + x2 = 0 + 4.642857143 Combine like terms: 0 + 4.642857143 = 4.642857143 0.8571428571x + x2 = 4.642857143 The x term is 0.8571428571x. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571x + 0.1836734694 + x2 = 4.642857143 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571x + x2 = 4.642857143 + 0.1836734694 Combine like terms: 4.642857143 + 0.1836734694 = 4.8265306124 0.1836734694 + 0.8571428571x + x2 = 4.8265306124 Factor a perfect square on the left side: (x + 0.4285714286)(x + 0.4285714286) = 4.8265306124 Calculate the square root of the right side: 2.196936643 Break this problem into two subproblems by setting (x + 0.4285714286) equal to 2.196936643 and -2.196936643.Subproblem 1
x + 0.4285714286 = 2.196936643 Simplifying x + 0.4285714286 = 2.196936643 Reorder the terms: 0.4285714286 + x = 2.196936643 Solving 0.4285714286 + x = 2.196936643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + x = 2.196936643 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = 2.196936643 + -0.4285714286 x = 2.196936643 + -0.4285714286 Combine like terms: 2.196936643 + -0.4285714286 = 1.7683652144 x = 1.7683652144 Simplifying x = 1.7683652144Subproblem 2
x + 0.4285714286 = -2.196936643 Simplifying x + 0.4285714286 = -2.196936643 Reorder the terms: 0.4285714286 + x = -2.196936643 Solving 0.4285714286 + x = -2.196936643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + x = -2.196936643 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + x = -2.196936643 + -0.4285714286 x = -2.196936643 + -0.4285714286 Combine like terms: -2.196936643 + -0.4285714286 = -2.6255080716 x = -2.6255080716 Simplifying x = -2.6255080716Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.7683652144, -2.6255080716}
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