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Simplifying 66 = (x + -4)(x + -0.5) Reorder the terms: 66 = (-4 + x)(x + -0.5) Reorder the terms: 66 = (-4 + x)(-0.5 + x) Multiply (-4 + x) * (-0.5 + x) 66 = (-4(-0.5 + x) + x(-0.5 + x)) 66 = ((-0.5 * -4 + x * -4) + x(-0.5 + x)) 66 = ((2 + -4x) + x(-0.5 + x)) 66 = (2 + -4x + (-0.5 * x + x * x)) 66 = (2 + -4x + (-0.5x + x2)) Combine like terms: -4x + -0.5x = -4.5x 66 = (2 + -4.5x + x2) Solving 66 = 2 + -4.5x + x2 Solving for variable 'x'. Combine like terms: 66 + -2 = 64 64 + 4.5x + -1x2 = 2 + -4.5x + x2 + -2 + 4.5x + -1x2 Reorder the terms: 64 + 4.5x + -1x2 = 2 + -2 + -4.5x + 4.5x + x2 + -1x2 Combine like terms: 2 + -2 = 0 64 + 4.5x + -1x2 = 0 + -4.5x + 4.5x + x2 + -1x2 64 + 4.5x + -1x2 = -4.5x + 4.5x + x2 + -1x2 Combine like terms: -4.5x + 4.5x = 0.0 64 + 4.5x + -1x2 = 0.0 + x2 + -1x2 64 + 4.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 64 + 4.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -64 + -4.5x + x2 = 0 Move the constant term to the right: Add '64' to each side of the equation. -64 + -4.5x + 64 + x2 = 0 + 64 Reorder the terms: -64 + 64 + -4.5x + x2 = 0 + 64 Combine like terms: -64 + 64 = 0 0 + -4.5x + x2 = 0 + 64 -4.5x + x2 = 0 + 64 Combine like terms: 0 + 64 = 64 -4.5x + x2 = 64 The x term is -4.5x. Take half its coefficient (-2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. -4.5x + 5.0625 + x2 = 64 + 5.0625 Reorder the terms: 5.0625 + -4.5x + x2 = 64 + 5.0625 Combine like terms: 64 + 5.0625 = 69.0625 5.0625 + -4.5x + x2 = 69.0625 Factor a perfect square on the left side: (x + -2.25)(x + -2.25) = 69.0625 Calculate the square root of the right side: 8.310385069 Break this problem into two subproblems by setting (x + -2.25) equal to 8.310385069 and -8.310385069.Subproblem 1
x + -2.25 = 8.310385069 Simplifying x + -2.25 = 8.310385069 Reorder the terms: -2.25 + x = 8.310385069 Solving -2.25 + x = 8.310385069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = 8.310385069 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = 8.310385069 + 2.25 x = 8.310385069 + 2.25 Combine like terms: 8.310385069 + 2.25 = 10.560385069 x = 10.560385069 Simplifying x = 10.560385069Subproblem 2
x + -2.25 = -8.310385069 Simplifying x + -2.25 = -8.310385069 Reorder the terms: -2.25 + x = -8.310385069 Solving -2.25 + x = -8.310385069 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + x = -8.310385069 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + x = -8.310385069 + 2.25 x = -8.310385069 + 2.25 Combine like terms: -8.310385069 + 2.25 = -6.060385069 x = -6.060385069 Simplifying x = -6.060385069Solution
The solution to the problem is based on the solutions from the subproblems. x = {10.560385069, -6.060385069}
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