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Simplifying (6x + -4)(7x + 2) = x Reorder the terms: (-4 + 6x)(7x + 2) = x Reorder the terms: (-4 + 6x)(2 + 7x) = x Multiply (-4 + 6x) * (2 + 7x) (-4(2 + 7x) + 6x * (2 + 7x)) = x ((2 * -4 + 7x * -4) + 6x * (2 + 7x)) = x ((-8 + -28x) + 6x * (2 + 7x)) = x (-8 + -28x + (2 * 6x + 7x * 6x)) = x (-8 + -28x + (12x + 42x2)) = x Combine like terms: -28x + 12x = -16x (-8 + -16x + 42x2) = x Solving -8 + -16x + 42x2 = x Solving for variable 'x'. Reorder the terms: -8 + -16x + -1x + 42x2 = x + -1x Combine like terms: -16x + -1x = -17x -8 + -17x + 42x2 = x + -1x Combine like terms: x + -1x = 0 -8 + -17x + 42x2 = 0 Begin completing the square. Divide all terms by 42 the coefficient of the squared term: Divide each side by '42'. -0.1904761905 + -0.4047619048x + x2 = 0 Move the constant term to the right: Add '0.1904761905' to each side of the equation. -0.1904761905 + -0.4047619048x + 0.1904761905 + x2 = 0 + 0.1904761905 Reorder the terms: -0.1904761905 + 0.1904761905 + -0.4047619048x + x2 = 0 + 0.1904761905 Combine like terms: -0.1904761905 + 0.1904761905 = 0.0000000000 0.0000000000 + -0.4047619048x + x2 = 0 + 0.1904761905 -0.4047619048x + x2 = 0 + 0.1904761905 Combine like terms: 0 + 0.1904761905 = 0.1904761905 -0.4047619048x + x2 = 0.1904761905 The x term is -0.4047619048x. Take half its coefficient (-0.2023809524). Square it (0.04095804989) and add it to both sides. Add '0.04095804989' to each side of the equation. -0.4047619048x + 0.04095804989 + x2 = 0.1904761905 + 0.04095804989 Reorder the terms: 0.04095804989 + -0.4047619048x + x2 = 0.1904761905 + 0.04095804989 Combine like terms: 0.1904761905 + 0.04095804989 = 0.23143424039 0.04095804989 + -0.4047619048x + x2 = 0.23143424039 Factor a perfect square on the left side: (x + -0.2023809524)(x + -0.2023809524) = 0.23143424039 Calculate the square root of the right side: 0.481076127 Break this problem into two subproblems by setting (x + -0.2023809524) equal to 0.481076127 and -0.481076127.Subproblem 1
x + -0.2023809524 = 0.481076127 Simplifying x + -0.2023809524 = 0.481076127 Reorder the terms: -0.2023809524 + x = 0.481076127 Solving -0.2023809524 + x = 0.481076127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2023809524' to each side of the equation. -0.2023809524 + 0.2023809524 + x = 0.481076127 + 0.2023809524 Combine like terms: -0.2023809524 + 0.2023809524 = 0.0000000000 0.0000000000 + x = 0.481076127 + 0.2023809524 x = 0.481076127 + 0.2023809524 Combine like terms: 0.481076127 + 0.2023809524 = 0.6834570794 x = 0.6834570794 Simplifying x = 0.6834570794Subproblem 2
x + -0.2023809524 = -0.481076127 Simplifying x + -0.2023809524 = -0.481076127 Reorder the terms: -0.2023809524 + x = -0.481076127 Solving -0.2023809524 + x = -0.481076127 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2023809524' to each side of the equation. -0.2023809524 + 0.2023809524 + x = -0.481076127 + 0.2023809524 Combine like terms: -0.2023809524 + 0.2023809524 = 0.0000000000 0.0000000000 + x = -0.481076127 + 0.2023809524 x = -0.481076127 + 0.2023809524 Combine like terms: -0.481076127 + 0.2023809524 = -0.2786951746 x = -0.2786951746 Simplifying x = -0.2786951746Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.6834570794, -0.2786951746}
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