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Simplifying (x + -3)(x + -2) = (x + 5)(2x + -2) + 21 Reorder the terms: (-3 + x)(x + -2) = (x + 5)(2x + -2) + 21 Reorder the terms: (-3 + x)(-2 + x) = (x + 5)(2x + -2) + 21 Multiply (-3 + x) * (-2 + x) (-3(-2 + x) + x(-2 + x)) = (x + 5)(2x + -2) + 21 ((-2 * -3 + x * -3) + x(-2 + x)) = (x + 5)(2x + -2) + 21 ((6 + -3x) + x(-2 + x)) = (x + 5)(2x + -2) + 21 (6 + -3x + (-2 * x + x * x)) = (x + 5)(2x + -2) + 21 (6 + -3x + (-2x + x2)) = (x + 5)(2x + -2) + 21 Combine like terms: -3x + -2x = -5x (6 + -5x + x2) = (x + 5)(2x + -2) + 21 Reorder the terms: 6 + -5x + x2 = (5 + x)(2x + -2) + 21 Reorder the terms: 6 + -5x + x2 = (5 + x)(-2 + 2x) + 21 Multiply (5 + x) * (-2 + 2x) 6 + -5x + x2 = (5(-2 + 2x) + x(-2 + 2x)) + 21 6 + -5x + x2 = ((-2 * 5 + 2x * 5) + x(-2 + 2x)) + 21 6 + -5x + x2 = ((-10 + 10x) + x(-2 + 2x)) + 21 6 + -5x + x2 = (-10 + 10x + (-2 * x + 2x * x)) + 21 6 + -5x + x2 = (-10 + 10x + (-2x + 2x2)) + 21 Combine like terms: 10x + -2x = 8x 6 + -5x + x2 = (-10 + 8x + 2x2) + 21 Reorder the terms: 6 + -5x + x2 = -10 + 21 + 8x + 2x2 Combine like terms: -10 + 21 = 11 6 + -5x + x2 = 11 + 8x + 2x2 Solving 6 + -5x + x2 = 11 + 8x + 2x2 Solving for variable 'x'. Reorder the terms: 6 + -11 + -5x + -8x + x2 + -2x2 = 11 + 8x + 2x2 + -11 + -8x + -2x2 Combine like terms: 6 + -11 = -5 -5 + -5x + -8x + x2 + -2x2 = 11 + 8x + 2x2 + -11 + -8x + -2x2 Combine like terms: -5x + -8x = -13x -5 + -13x + x2 + -2x2 = 11 + 8x + 2x2 + -11 + -8x + -2x2 Combine like terms: x2 + -2x2 = -1x2 -5 + -13x + -1x2 = 11 + 8x + 2x2 + -11 + -8x + -2x2 Reorder the terms: -5 + -13x + -1x2 = 11 + -11 + 8x + -8x + 2x2 + -2x2 Combine like terms: 11 + -11 = 0 -5 + -13x + -1x2 = 0 + 8x + -8x + 2x2 + -2x2 -5 + -13x + -1x2 = 8x + -8x + 2x2 + -2x2 Combine like terms: 8x + -8x = 0 -5 + -13x + -1x2 = 0 + 2x2 + -2x2 -5 + -13x + -1x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 -5 + -13x + -1x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(5 + 13x + x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(5 + 13x + x2)' equal to zero and attempt to solve: Simplifying 5 + 13x + x2 = 0 Solving 5 + 13x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 13x + -5 + x2 = 0 + -5 Reorder the terms: 5 + -5 + 13x + x2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 13x + x2 = 0 + -5 13x + x2 = 0 + -5 Combine like terms: 0 + -5 = -5 13x + x2 = -5 The x term is 13x. Take half its coefficient (6.5). Square it (42.25) and add it to both sides. Add '42.25' to each side of the equation. 13x + 42.25 + x2 = -5 + 42.25 Reorder the terms: 42.25 + 13x + x2 = -5 + 42.25 Combine like terms: -5 + 42.25 = 37.25 42.25 + 13x + x2 = 37.25 Factor a perfect square on the left side: (x + 6.5)(x + 6.5) = 37.25 Calculate the square root of the right side: 6.103277808 Break this problem into two subproblems by setting (x + 6.5) equal to 6.103277808 and -6.103277808.Subproblem 1
x + 6.5 = 6.103277808 Simplifying x + 6.5 = 6.103277808 Reorder the terms: 6.5 + x = 6.103277808 Solving 6.5 + x = 6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = 6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = 6.103277808 + -6.5 x = 6.103277808 + -6.5 Combine like terms: 6.103277808 + -6.5 = -0.396722192 x = -0.396722192 Simplifying x = -0.396722192Subproblem 2
x + 6.5 = -6.103277808 Simplifying x + 6.5 = -6.103277808 Reorder the terms: 6.5 + x = -6.103277808 Solving 6.5 + x = -6.103277808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + x = -6.103277808 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + x = -6.103277808 + -6.5 x = -6.103277808 + -6.5 Combine like terms: -6.103277808 + -6.5 = -12.603277808 x = -12.603277808 Simplifying x = -12.603277808Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.396722192, -12.603277808}Solution
x = {-0.396722192, -12.603277808}
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