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Simplifying 6(x + 24)(x + -6) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) Reorder the terms: 6(24 + x)(x + -6) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) Reorder the terms: 6(24 + x)(-6 + x) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) Multiply (24 + x) * (-6 + x) 6(24(-6 + x) + x(-6 + x)) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) 6((-6 * 24 + x * 24) + x(-6 + x)) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) 6((-144 + 24x) + x(-6 + x)) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) 6(-144 + 24x + (-6 * x + x * x)) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) 6(-144 + 24x + (-6x + x2)) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) Combine like terms: 24x + -6x = 18x 6(-144 + 18x + x2) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) (-144 * 6 + 18x * 6 + x2 * 6) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) (-864 + 108x + 6x2) + 3(x + 12)(x + -18) = 19(x + 45) + -1(x + -36) Reorder the terms: -864 + 108x + 6x2 + 3(12 + x)(x + -18) = 19(x + 45) + -1(x + -36) Reorder the terms: -864 + 108x + 6x2 + 3(12 + x)(-18 + x) = 19(x + 45) + -1(x + -36) Multiply (12 + x) * (-18 + x) -864 + 108x + 6x2 + 3(12(-18 + x) + x(-18 + x)) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + 3((-18 * 12 + x * 12) + x(-18 + x)) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + 3((-216 + 12x) + x(-18 + x)) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + 3(-216 + 12x + (-18 * x + x * x)) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + 3(-216 + 12x + (-18x + x2)) = 19(x + 45) + -1(x + -36) Combine like terms: 12x + -18x = -6x -864 + 108x + 6x2 + 3(-216 + -6x + x2) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + (-216 * 3 + -6x * 3 + x2 * 3) = 19(x + 45) + -1(x + -36) -864 + 108x + 6x2 + (-648 + -18x + 3x2) = 19(x + 45) + -1(x + -36) Reorder the terms: -864 + -648 + 108x + -18x + 6x2 + 3x2 = 19(x + 45) + -1(x + -36) Combine like terms: -864 + -648 = -1512 -1512 + 108x + -18x + 6x2 + 3x2 = 19(x + 45) + -1(x + -36) Combine like terms: 108x + -18x = 90x -1512 + 90x + 6x2 + 3x2 = 19(x + 45) + -1(x + -36) Combine like terms: 6x2 + 3x2 = 9x2 -1512 + 90x + 9x2 = 19(x + 45) + -1(x + -36) Reorder the terms: -1512 + 90x + 9x2 = 19(45 + x) + -1(x + -36) -1512 + 90x + 9x2 = (45 * 19 + x * 19) + -1(x + -36) -1512 + 90x + 9x2 = (855 + 19x) + -1(x + -36) Reorder the terms: -1512 + 90x + 9x2 = 855 + 19x + -1(-36 + x) -1512 + 90x + 9x2 = 855 + 19x + (-36 * -1 + x * -1) -1512 + 90x + 9x2 = 855 + 19x + (36 + -1x) Reorder the terms: -1512 + 90x + 9x2 = 855 + 36 + 19x + -1x Combine like terms: 855 + 36 = 891 -1512 + 90x + 9x2 = 891 + 19x + -1x Combine like terms: 19x + -1x = 18x -1512 + 90x + 9x2 = 891 + 18x Solving -1512 + 90x + 9x2 = 891 + 18x Solving for variable 'x'. Reorder the terms: -1512 + -891 + 90x + -18x + 9x2 = 891 + 18x + -891 + -18x Combine like terms: -1512 + -891 = -2403 -2403 + 90x + -18x + 9x2 = 891 + 18x + -891 + -18x Combine like terms: 90x + -18x = 72x -2403 + 72x + 9x2 = 891 + 18x + -891 + -18x Reorder the terms: -2403 + 72x + 9x2 = 891 + -891 + 18x + -18x Combine like terms: 891 + -891 = 0 -2403 + 72x + 9x2 = 0 + 18x + -18x -2403 + 72x + 9x2 = 18x + -18x Combine like terms: 18x + -18x = 0 -2403 + 72x + 9x2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-267 + 8x + x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-267 + 8x + x2)' equal to zero and attempt to solve: Simplifying -267 + 8x + x2 = 0 Solving -267 + 8x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '267' to each side of the equation. -267 + 8x + 267 + x2 = 0 + 267 Reorder the terms: -267 + 267 + 8x + x2 = 0 + 267 Combine like terms: -267 + 267 = 0 0 + 8x + x2 = 0 + 267 8x + x2 = 0 + 267 Combine like terms: 0 + 267 = 267 8x + x2 = 267 The x term is 8x. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8x + 16 + x2 = 267 + 16 Reorder the terms: 16 + 8x + x2 = 267 + 16 Combine like terms: 267 + 16 = 283 16 + 8x + x2 = 283 Factor a perfect square on the left side: (x + 4)(x + 4) = 283 Calculate the square root of the right side: 16.822603841 Break this problem into two subproblems by setting (x + 4) equal to 16.822603841 and -16.822603841.Subproblem 1
x + 4 = 16.822603841 Simplifying x + 4 = 16.822603841 Reorder the terms: 4 + x = 16.822603841 Solving 4 + x = 16.822603841 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = 16.822603841 + -4 Combine like terms: 4 + -4 = 0 0 + x = 16.822603841 + -4 x = 16.822603841 + -4 Combine like terms: 16.822603841 + -4 = 12.822603841 x = 12.822603841 Simplifying x = 12.822603841Subproblem 2
x + 4 = -16.822603841 Simplifying x + 4 = -16.822603841 Reorder the terms: 4 + x = -16.822603841 Solving 4 + x = -16.822603841 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + x = -16.822603841 + -4 Combine like terms: 4 + -4 = 0 0 + x = -16.822603841 + -4 x = -16.822603841 + -4 Combine like terms: -16.822603841 + -4 = -20.822603841 x = -20.822603841 Simplifying x = -20.822603841Solution
The solution to the problem is based on the solutions from the subproblems. x = {12.822603841, -20.822603841}Solution
x = {12.822603841, -20.822603841}
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