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Simplifying 420x2 + -1120x + 640 = 0 Reorder the terms: 640 + -1120x + 420x2 = 0 Solving 640 + -1120x + 420x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '20'. 20(32 + -56x + 21x2) = 0 Ignore the factor 20.Subproblem 1
Set the factor '(32 + -56x + 21x2)' equal to zero and attempt to solve: Simplifying 32 + -56x + 21x2 = 0 Solving 32 + -56x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. 1.523809524 + -2.666666667x + x2 = 0 Move the constant term to the right: Add '-1.523809524' to each side of the equation. 1.523809524 + -2.666666667x + -1.523809524 + x2 = 0 + -1.523809524 Reorder the terms: 1.523809524 + -1.523809524 + -2.666666667x + x2 = 0 + -1.523809524 Combine like terms: 1.523809524 + -1.523809524 = 0.000000000 0.000000000 + -2.666666667x + x2 = 0 + -1.523809524 -2.666666667x + x2 = 0 + -1.523809524 Combine like terms: 0 + -1.523809524 = -1.523809524 -2.666666667x + x2 = -1.523809524 The x term is -2.666666667x. Take half its coefficient (-1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. -2.666666667x + 1.777777780 + x2 = -1.523809524 + 1.777777780 Reorder the terms: 1.777777780 + -2.666666667x + x2 = -1.523809524 + 1.777777780 Combine like terms: -1.523809524 + 1.777777780 = 0.253968256 1.777777780 + -2.666666667x + x2 = 0.253968256 Factor a perfect square on the left side: (x + -1.333333334)(x + -1.333333334) = 0.253968256 Calculate the square root of the right side: 0.503952633 Break this problem into two subproblems by setting (x + -1.333333334) equal to 0.503952633 and -0.503952633.Subproblem 1
x + -1.333333334 = 0.503952633 Simplifying x + -1.333333334 = 0.503952633 Reorder the terms: -1.333333334 + x = 0.503952633 Solving -1.333333334 + x = 0.503952633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = 0.503952633 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = 0.503952633 + 1.333333334 x = 0.503952633 + 1.333333334 Combine like terms: 0.503952633 + 1.333333334 = 1.837285967 x = 1.837285967 Simplifying x = 1.837285967Subproblem 2
x + -1.333333334 = -0.503952633 Simplifying x + -1.333333334 = -0.503952633 Reorder the terms: -1.333333334 + x = -0.503952633 Solving -1.333333334 + x = -0.503952633 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.333333334' to each side of the equation. -1.333333334 + 1.333333334 + x = -0.503952633 + 1.333333334 Combine like terms: -1.333333334 + 1.333333334 = 0.000000000 0.000000000 + x = -0.503952633 + 1.333333334 x = -0.503952633 + 1.333333334 Combine like terms: -0.503952633 + 1.333333334 = 0.829380701 x = 0.829380701 Simplifying x = 0.829380701Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.837285967, 0.829380701}Solution
x = {1.837285967, 0.829380701}
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