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Simplifying (3x + -1)(3x + -1) + -1(5x + -3)(5x + -3) = -1(4x + -3) Reorder the terms: (-1 + 3x)(3x + -1) + -1(5x + -3)(5x + -3) = -1(4x + -3) Reorder the terms: (-1 + 3x)(-1 + 3x) + -1(5x + -3)(5x + -3) = -1(4x + -3) Multiply (-1 + 3x) * (-1 + 3x) (-1(-1 + 3x) + 3x * (-1 + 3x)) + -1(5x + -3)(5x + -3) = -1(4x + -3) ((-1 * -1 + 3x * -1) + 3x * (-1 + 3x)) + -1(5x + -3)(5x + -3) = -1(4x + -3) ((1 + -3x) + 3x * (-1 + 3x)) + -1(5x + -3)(5x + -3) = -1(4x + -3) (1 + -3x + (-1 * 3x + 3x * 3x)) + -1(5x + -3)(5x + -3) = -1(4x + -3) (1 + -3x + (-3x + 9x2)) + -1(5x + -3)(5x + -3) = -1(4x + -3) Combine like terms: -3x + -3x = -6x (1 + -6x + 9x2) + -1(5x + -3)(5x + -3) = -1(4x + -3) Reorder the terms: 1 + -6x + 9x2 + -1(-3 + 5x)(5x + -3) = -1(4x + -3) Reorder the terms: 1 + -6x + 9x2 + -1(-3 + 5x)(-3 + 5x) = -1(4x + -3) Multiply (-3 + 5x) * (-3 + 5x) 1 + -6x + 9x2 + -1(-3(-3 + 5x) + 5x * (-3 + 5x)) = -1(4x + -3) 1 + -6x + 9x2 + -1((-3 * -3 + 5x * -3) + 5x * (-3 + 5x)) = -1(4x + -3) 1 + -6x + 9x2 + -1((9 + -15x) + 5x * (-3 + 5x)) = -1(4x + -3) 1 + -6x + 9x2 + -1(9 + -15x + (-3 * 5x + 5x * 5x)) = -1(4x + -3) 1 + -6x + 9x2 + -1(9 + -15x + (-15x + 25x2)) = -1(4x + -3) Combine like terms: -15x + -15x = -30x 1 + -6x + 9x2 + -1(9 + -30x + 25x2) = -1(4x + -3) 1 + -6x + 9x2 + (9 * -1 + -30x * -1 + 25x2 * -1) = -1(4x + -3) 1 + -6x + 9x2 + (-9 + 30x + -25x2) = -1(4x + -3) Reorder the terms: 1 + -9 + -6x + 30x + 9x2 + -25x2 = -1(4x + -3) Combine like terms: 1 + -9 = -8 -8 + -6x + 30x + 9x2 + -25x2 = -1(4x + -3) Combine like terms: -6x + 30x = 24x -8 + 24x + 9x2 + -25x2 = -1(4x + -3) Combine like terms: 9x2 + -25x2 = -16x2 -8 + 24x + -16x2 = -1(4x + -3) Reorder the terms: -8 + 24x + -16x2 = -1(-3 + 4x) -8 + 24x + -16x2 = (-3 * -1 + 4x * -1) -8 + 24x + -16x2 = (3 + -4x) Solving -8 + 24x + -16x2 = 3 + -4x Solving for variable 'x'. Reorder the terms: -8 + -3 + 24x + 4x + -16x2 = 3 + -4x + -3 + 4x Combine like terms: -8 + -3 = -11 -11 + 24x + 4x + -16x2 = 3 + -4x + -3 + 4x Combine like terms: 24x + 4x = 28x -11 + 28x + -16x2 = 3 + -4x + -3 + 4x Reorder the terms: -11 + 28x + -16x2 = 3 + -3 + -4x + 4x Combine like terms: 3 + -3 = 0 -11 + 28x + -16x2 = 0 + -4x + 4x -11 + 28x + -16x2 = -4x + 4x Combine like terms: -4x + 4x = 0 -11 + 28x + -16x2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. 0.6875 + -1.75x + x2 = 0 Move the constant term to the right: Add '-0.6875' to each side of the equation. 0.6875 + -1.75x + -0.6875 + x2 = 0 + -0.6875 Reorder the terms: 0.6875 + -0.6875 + -1.75x + x2 = 0 + -0.6875 Combine like terms: 0.6875 + -0.6875 = 0.0000 0.0000 + -1.75x + x2 = 0 + -0.6875 -1.75x + x2 = 0 + -0.6875 Combine like terms: 0 + -0.6875 = -0.6875 -1.75x + x2 = -0.6875 The x term is -1.75x. Take half its coefficient (-0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. -1.75x + 0.765625 + x2 = -0.6875 + 0.765625 Reorder the terms: 0.765625 + -1.75x + x2 = -0.6875 + 0.765625 Combine like terms: -0.6875 + 0.765625 = 0.078125 0.765625 + -1.75x + x2 = 0.078125 Factor a perfect square on the left side: (x + -0.875)(x + -0.875) = 0.078125 Calculate the square root of the right side: 0.279508497 Break this problem into two subproblems by setting (x + -0.875) equal to 0.279508497 and -0.279508497.Subproblem 1
x + -0.875 = 0.279508497 Simplifying x + -0.875 = 0.279508497 Reorder the terms: -0.875 + x = 0.279508497 Solving -0.875 + x = 0.279508497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = 0.279508497 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = 0.279508497 + 0.875 x = 0.279508497 + 0.875 Combine like terms: 0.279508497 + 0.875 = 1.154508497 x = 1.154508497 Simplifying x = 1.154508497Subproblem 2
x + -0.875 = -0.279508497 Simplifying x + -0.875 = -0.279508497 Reorder the terms: -0.875 + x = -0.279508497 Solving -0.875 + x = -0.279508497 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.875' to each side of the equation. -0.875 + 0.875 + x = -0.279508497 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + x = -0.279508497 + 0.875 x = -0.279508497 + 0.875 Combine like terms: -0.279508497 + 0.875 = 0.595491503 x = 0.595491503 Simplifying x = 0.595491503Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.154508497, 0.595491503}
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