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Simplifying (4x + -16)(4x + -16) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) Reorder the terms: (-16 + 4x)(4x + -16) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) Reorder the terms: (-16 + 4x)(-16 + 4x) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) Multiply (-16 + 4x) * (-16 + 4x) (-16(-16 + 4x) + 4x * (-16 + 4x)) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) ((-16 * -16 + 4x * -16) + 4x * (-16 + 4x)) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) ((256 + -64x) + 4x * (-16 + 4x)) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) (256 + -64x + (-16 * 4x + 4x * 4x)) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) (256 + -64x + (-64x + 16x2)) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) Combine like terms: -64x + -64x = -128x (256 + -128x + 16x2) + -1(x + -2)(x + -2) = (8 + -1x)(8 + -1x) Reorder the terms: 256 + -128x + 16x2 + -1(-2 + x)(x + -2) = (8 + -1x)(8 + -1x) Reorder the terms: 256 + -128x + 16x2 + -1(-2 + x)(-2 + x) = (8 + -1x)(8 + -1x) Multiply (-2 + x) * (-2 + x) 256 + -128x + 16x2 + -1(-2(-2 + x) + x(-2 + x)) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + -1((-2 * -2 + x * -2) + x(-2 + x)) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + -1((4 + -2x) + x(-2 + x)) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + -1(4 + -2x + (-2 * x + x * x)) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + -1(4 + -2x + (-2x + x2)) = (8 + -1x)(8 + -1x) Combine like terms: -2x + -2x = -4x 256 + -128x + 16x2 + -1(4 + -4x + x2) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + (4 * -1 + -4x * -1 + x2 * -1) = (8 + -1x)(8 + -1x) 256 + -128x + 16x2 + (-4 + 4x + -1x2) = (8 + -1x)(8 + -1x) Reorder the terms: 256 + -4 + -128x + 4x + 16x2 + -1x2 = (8 + -1x)(8 + -1x) Combine like terms: 256 + -4 = 252 252 + -128x + 4x + 16x2 + -1x2 = (8 + -1x)(8 + -1x) Combine like terms: -128x + 4x = -124x 252 + -124x + 16x2 + -1x2 = (8 + -1x)(8 + -1x) Combine like terms: 16x2 + -1x2 = 15x2 252 + -124x + 15x2 = (8 + -1x)(8 + -1x) Multiply (8 + -1x) * (8 + -1x) 252 + -124x + 15x2 = (8(8 + -1x) + -1x * (8 + -1x)) 252 + -124x + 15x2 = ((8 * 8 + -1x * 8) + -1x * (8 + -1x)) 252 + -124x + 15x2 = ((64 + -8x) + -1x * (8 + -1x)) 252 + -124x + 15x2 = (64 + -8x + (8 * -1x + -1x * -1x)) 252 + -124x + 15x2 = (64 + -8x + (-8x + 1x2)) Combine like terms: -8x + -8x = -16x 252 + -124x + 15x2 = (64 + -16x + 1x2) Solving 252 + -124x + 15x2 = 64 + -16x + 1x2 Solving for variable 'x'. Reorder the terms: 252 + -64 + -124x + 16x + 15x2 + -1x2 = 64 + -16x + 1x2 + -64 + 16x + -1x2 Combine like terms: 252 + -64 = 188 188 + -124x + 16x + 15x2 + -1x2 = 64 + -16x + 1x2 + -64 + 16x + -1x2 Combine like terms: -124x + 16x = -108x 188 + -108x + 15x2 + -1x2 = 64 + -16x + 1x2 + -64 + 16x + -1x2 Combine like terms: 15x2 + -1x2 = 14x2 188 + -108x + 14x2 = 64 + -16x + 1x2 + -64 + 16x + -1x2 Reorder the terms: 188 + -108x + 14x2 = 64 + -64 + -16x + 16x + 1x2 + -1x2 Combine like terms: 64 + -64 = 0 188 + -108x + 14x2 = 0 + -16x + 16x + 1x2 + -1x2 188 + -108x + 14x2 = -16x + 16x + 1x2 + -1x2 Combine like terms: -16x + 16x = 0 188 + -108x + 14x2 = 0 + 1x2 + -1x2 188 + -108x + 14x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 188 + -108x + 14x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(94 + -54x + 7x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(94 + -54x + 7x2)' equal to zero and attempt to solve: Simplifying 94 + -54x + 7x2 = 0 Solving 94 + -54x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 13.42857143 + -7.714285714x + x2 = 0 Move the constant term to the right: Add '-13.42857143' to each side of the equation. 13.42857143 + -7.714285714x + -13.42857143 + x2 = 0 + -13.42857143 Reorder the terms: 13.42857143 + -13.42857143 + -7.714285714x + x2 = 0 + -13.42857143 Combine like terms: 13.42857143 + -13.42857143 = 0.00000000 0.00000000 + -7.714285714x + x2 = 0 + -13.42857143 -7.714285714x + x2 = 0 + -13.42857143 Combine like terms: 0 + -13.42857143 = -13.42857143 -7.714285714x + x2 = -13.42857143 The x term is -7.714285714x. Take half its coefficient (-3.857142857). Square it (14.87755102) and add it to both sides. Add '14.87755102' to each side of the equation. -7.714285714x + 14.87755102 + x2 = -13.42857143 + 14.87755102 Reorder the terms: 14.87755102 + -7.714285714x + x2 = -13.42857143 + 14.87755102 Combine like terms: -13.42857143 + 14.87755102 = 1.44897959 14.87755102 + -7.714285714x + x2 = 1.44897959 Factor a perfect square on the left side: (x + -3.857142857)(x + -3.857142857) = 1.44897959 Calculate the square root of the right side: 1.203735681 Break this problem into two subproblems by setting (x + -3.857142857) equal to 1.203735681 and -1.203735681.Subproblem 1
x + -3.857142857 = 1.203735681 Simplifying x + -3.857142857 = 1.203735681 Reorder the terms: -3.857142857 + x = 1.203735681 Solving -3.857142857 + x = 1.203735681 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.857142857' to each side of the equation. -3.857142857 + 3.857142857 + x = 1.203735681 + 3.857142857 Combine like terms: -3.857142857 + 3.857142857 = 0.000000000 0.000000000 + x = 1.203735681 + 3.857142857 x = 1.203735681 + 3.857142857 Combine like terms: 1.203735681 + 3.857142857 = 5.060878538 x = 5.060878538 Simplifying x = 5.060878538Subproblem 2
x + -3.857142857 = -1.203735681 Simplifying x + -3.857142857 = -1.203735681 Reorder the terms: -3.857142857 + x = -1.203735681 Solving -3.857142857 + x = -1.203735681 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.857142857' to each side of the equation. -3.857142857 + 3.857142857 + x = -1.203735681 + 3.857142857 Combine like terms: -3.857142857 + 3.857142857 = 0.000000000 0.000000000 + x = -1.203735681 + 3.857142857 x = -1.203735681 + 3.857142857 Combine like terms: -1.203735681 + 3.857142857 = 2.653407176 x = 2.653407176 Simplifying x = 2.653407176Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.060878538, 2.653407176}Solution
x = {5.060878538, 2.653407176}
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