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Simplifying 8(2x + 6) * 8x = 12x + 15 Reorder the terms: 8(6 + 2x) * 8x = 12x + 15 Reorder the terms for easier multiplication: 8 * 8x(6 + 2x) = 12x + 15 Multiply 8 * 8 64x(6 + 2x) = 12x + 15 (6 * 64x + 2x * 64x) = 12x + 15 (384x + 128x2) = 12x + 15 Reorder the terms: 384x + 128x2 = 15 + 12x Solving 384x + 128x2 = 15 + 12x Solving for variable 'x'. Reorder the terms: -15 + 384x + -12x + 128x2 = 15 + 12x + -15 + -12x Combine like terms: 384x + -12x = 372x -15 + 372x + 128x2 = 15 + 12x + -15 + -12x Reorder the terms: -15 + 372x + 128x2 = 15 + -15 + 12x + -12x Combine like terms: 15 + -15 = 0 -15 + 372x + 128x2 = 0 + 12x + -12x -15 + 372x + 128x2 = 12x + -12x Combine like terms: 12x + -12x = 0 -15 + 372x + 128x2 = 0 Begin completing the square. Divide all terms by 128 the coefficient of the squared term: Divide each side by '128'. -0.1171875 + 2.90625x + x2 = 0 Move the constant term to the right: Add '0.1171875' to each side of the equation. -0.1171875 + 2.90625x + 0.1171875 + x2 = 0 + 0.1171875 Reorder the terms: -0.1171875 + 0.1171875 + 2.90625x + x2 = 0 + 0.1171875 Combine like terms: -0.1171875 + 0.1171875 = 0.0000000 0.0000000 + 2.90625x + x2 = 0 + 0.1171875 2.90625x + x2 = 0 + 0.1171875 Combine like terms: 0 + 0.1171875 = 0.1171875 2.90625x + x2 = 0.1171875 The x term is 2.90625x. Take half its coefficient (1.453125). Square it (2.111572266) and add it to both sides. Add '2.111572266' to each side of the equation. 2.90625x + 2.111572266 + x2 = 0.1171875 + 2.111572266 Reorder the terms: 2.111572266 + 2.90625x + x2 = 0.1171875 + 2.111572266 Combine like terms: 0.1171875 + 2.111572266 = 2.228759766 2.111572266 + 2.90625x + x2 = 2.228759766 Factor a perfect square on the left side: (x + 1.453125)(x + 1.453125) = 2.228759766 Calculate the square root of the right side: 1.492903133 Break this problem into two subproblems by setting (x + 1.453125) equal to 1.492903133 and -1.492903133.Subproblem 1
x + 1.453125 = 1.492903133 Simplifying x + 1.453125 = 1.492903133 Reorder the terms: 1.453125 + x = 1.492903133 Solving 1.453125 + x = 1.492903133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.453125' to each side of the equation. 1.453125 + -1.453125 + x = 1.492903133 + -1.453125 Combine like terms: 1.453125 + -1.453125 = 0.000000 0.000000 + x = 1.492903133 + -1.453125 x = 1.492903133 + -1.453125 Combine like terms: 1.492903133 + -1.453125 = 0.039778133 x = 0.039778133 Simplifying x = 0.039778133Subproblem 2
x + 1.453125 = -1.492903133 Simplifying x + 1.453125 = -1.492903133 Reorder the terms: 1.453125 + x = -1.492903133 Solving 1.453125 + x = -1.492903133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.453125' to each side of the equation. 1.453125 + -1.453125 + x = -1.492903133 + -1.453125 Combine like terms: 1.453125 + -1.453125 = 0.000000 0.000000 + x = -1.492903133 + -1.453125 x = -1.492903133 + -1.453125 Combine like terms: -1.492903133 + -1.453125 = -2.946028133 x = -2.946028133 Simplifying x = -2.946028133Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.039778133, -2.946028133}
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