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Simplifying 47x + -22 = 7(6x + -3) * 5x + -1 Reorder the terms: -22 + 47x = 7(6x + -3) * 5x + -1 Reorder the terms: -22 + 47x = 7(-3 + 6x) * 5x + -1 Reorder the terms for easier multiplication: -22 + 47x = 7 * 5x(-3 + 6x) + -1 Multiply 7 * 5 -22 + 47x = 35x(-3 + 6x) + -1 -22 + 47x = (-3 * 35x + 6x * 35x) + -1 -22 + 47x = (-105x + 210x2) + -1 Reorder the terms: -22 + 47x = -1 + -105x + 210x2 Solving -22 + 47x = -1 + -105x + 210x2 Solving for variable 'x'. Reorder the terms: -22 + 1 + 47x + 105x + -210x2 = -1 + -105x + 210x2 + 1 + 105x + -210x2 Combine like terms: -22 + 1 = -21 -21 + 47x + 105x + -210x2 = -1 + -105x + 210x2 + 1 + 105x + -210x2 Combine like terms: 47x + 105x = 152x -21 + 152x + -210x2 = -1 + -105x + 210x2 + 1 + 105x + -210x2 Reorder the terms: -21 + 152x + -210x2 = -1 + 1 + -105x + 105x + 210x2 + -210x2 Combine like terms: -1 + 1 = 0 -21 + 152x + -210x2 = 0 + -105x + 105x + 210x2 + -210x2 -21 + 152x + -210x2 = -105x + 105x + 210x2 + -210x2 Combine like terms: -105x + 105x = 0 -21 + 152x + -210x2 = 0 + 210x2 + -210x2 -21 + 152x + -210x2 = 210x2 + -210x2 Combine like terms: 210x2 + -210x2 = 0 -21 + 152x + -210x2 = 0 Begin completing the square. Divide all terms by -210 the coefficient of the squared term: Divide each side by '-210'. 0.1 + -0.7238095238x + x2 = 0 Move the constant term to the right: Add '-0.1' to each side of the equation. 0.1 + -0.7238095238x + -0.1 + x2 = 0 + -0.1 Reorder the terms: 0.1 + -0.1 + -0.7238095238x + x2 = 0 + -0.1 Combine like terms: 0.1 + -0.1 = 0.0 0.0 + -0.7238095238x + x2 = 0 + -0.1 -0.7238095238x + x2 = 0 + -0.1 Combine like terms: 0 + -0.1 = -0.1 -0.7238095238x + x2 = -0.1 The x term is -0.7238095238x. Take half its coefficient (-0.3619047619). Square it (0.1309750567) and add it to both sides. Add '0.1309750567' to each side of the equation. -0.7238095238x + 0.1309750567 + x2 = -0.1 + 0.1309750567 Reorder the terms: 0.1309750567 + -0.7238095238x + x2 = -0.1 + 0.1309750567 Combine like terms: -0.1 + 0.1309750567 = 0.0309750567 0.1309750567 + -0.7238095238x + x2 = 0.0309750567 Factor a perfect square on the left side: (x + -0.3619047619)(x + -0.3619047619) = 0.0309750567 Calculate the square root of the right side: 0.17599732 Break this problem into two subproblems by setting (x + -0.3619047619) equal to 0.17599732 and -0.17599732.Subproblem 1
x + -0.3619047619 = 0.17599732 Simplifying x + -0.3619047619 = 0.17599732 Reorder the terms: -0.3619047619 + x = 0.17599732 Solving -0.3619047619 + x = 0.17599732 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3619047619' to each side of the equation. -0.3619047619 + 0.3619047619 + x = 0.17599732 + 0.3619047619 Combine like terms: -0.3619047619 + 0.3619047619 = 0.0000000000 0.0000000000 + x = 0.17599732 + 0.3619047619 x = 0.17599732 + 0.3619047619 Combine like terms: 0.17599732 + 0.3619047619 = 0.5379020819 x = 0.5379020819 Simplifying x = 0.5379020819Subproblem 2
x + -0.3619047619 = -0.17599732 Simplifying x + -0.3619047619 = -0.17599732 Reorder the terms: -0.3619047619 + x = -0.17599732 Solving -0.3619047619 + x = -0.17599732 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3619047619' to each side of the equation. -0.3619047619 + 0.3619047619 + x = -0.17599732 + 0.3619047619 Combine like terms: -0.3619047619 + 0.3619047619 = 0.0000000000 0.0000000000 + x = -0.17599732 + 0.3619047619 x = -0.17599732 + 0.3619047619 Combine like terms: -0.17599732 + 0.3619047619 = 0.1859074419 x = 0.1859074419 Simplifying x = 0.1859074419Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5379020819, 0.1859074419}
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