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Simplifying (3x + -2)(5x + -7) = x Reorder the terms: (-2 + 3x)(5x + -7) = x Reorder the terms: (-2 + 3x)(-7 + 5x) = x Multiply (-2 + 3x) * (-7 + 5x) (-2(-7 + 5x) + 3x * (-7 + 5x)) = x ((-7 * -2 + 5x * -2) + 3x * (-7 + 5x)) = x ((14 + -10x) + 3x * (-7 + 5x)) = x (14 + -10x + (-7 * 3x + 5x * 3x)) = x (14 + -10x + (-21x + 15x2)) = x Combine like terms: -10x + -21x = -31x (14 + -31x + 15x2) = x Solving 14 + -31x + 15x2 = x Solving for variable 'x'. Reorder the terms: 14 + -31x + -1x + 15x2 = x + -1x Combine like terms: -31x + -1x = -32x 14 + -32x + 15x2 = x + -1x Combine like terms: x + -1x = 0 14 + -32x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. 0.9333333333 + -2.133333333x + x2 = 0 Move the constant term to the right: Add '-0.9333333333' to each side of the equation. 0.9333333333 + -2.133333333x + -0.9333333333 + x2 = 0 + -0.9333333333 Reorder the terms: 0.9333333333 + -0.9333333333 + -2.133333333x + x2 = 0 + -0.9333333333 Combine like terms: 0.9333333333 + -0.9333333333 = 0.0000000000 0.0000000000 + -2.133333333x + x2 = 0 + -0.9333333333 -2.133333333x + x2 = 0 + -0.9333333333 Combine like terms: 0 + -0.9333333333 = -0.9333333333 -2.133333333x + x2 = -0.9333333333 The x term is -2.133333333x. Take half its coefficient (-1.066666667). Square it (1.137777778) and add it to both sides. Add '1.137777778' to each side of the equation. -2.133333333x + 1.137777778 + x2 = -0.9333333333 + 1.137777778 Reorder the terms: 1.137777778 + -2.133333333x + x2 = -0.9333333333 + 1.137777778 Combine like terms: -0.9333333333 + 1.137777778 = 0.2044444447 1.137777778 + -2.133333333x + x2 = 0.2044444447 Factor a perfect square on the left side: (x + -1.066666667)(x + -1.066666667) = 0.2044444447 Calculate the square root of the right side: 0.452155332 Break this problem into two subproblems by setting (x + -1.066666667) equal to 0.452155332 and -0.452155332.Subproblem 1
x + -1.066666667 = 0.452155332 Simplifying x + -1.066666667 = 0.452155332 Reorder the terms: -1.066666667 + x = 0.452155332 Solving -1.066666667 + x = 0.452155332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + x = 0.452155332 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + x = 0.452155332 + 1.066666667 x = 0.452155332 + 1.066666667 Combine like terms: 0.452155332 + 1.066666667 = 1.518821999 x = 1.518821999 Simplifying x = 1.518821999Subproblem 2
x + -1.066666667 = -0.452155332 Simplifying x + -1.066666667 = -0.452155332 Reorder the terms: -1.066666667 + x = -0.452155332 Solving -1.066666667 + x = -0.452155332 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.066666667' to each side of the equation. -1.066666667 + 1.066666667 + x = -0.452155332 + 1.066666667 Combine like terms: -1.066666667 + 1.066666667 = 0.000000000 0.000000000 + x = -0.452155332 + 1.066666667 x = -0.452155332 + 1.066666667 Combine like terms: -0.452155332 + 1.066666667 = 0.614511335 x = 0.614511335 Simplifying x = 0.614511335Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.518821999, 0.614511335}
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