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Simplifying (3x + 18)(9x + -27) = 99 Reorder the terms: (18 + 3x)(9x + -27) = 99 Reorder the terms: (18 + 3x)(-27 + 9x) = 99 Multiply (18 + 3x) * (-27 + 9x) (18(-27 + 9x) + 3x * (-27 + 9x)) = 99 ((-27 * 18 + 9x * 18) + 3x * (-27 + 9x)) = 99 ((-486 + 162x) + 3x * (-27 + 9x)) = 99 (-486 + 162x + (-27 * 3x + 9x * 3x)) = 99 (-486 + 162x + (-81x + 27x2)) = 99 Combine like terms: 162x + -81x = 81x (-486 + 81x + 27x2) = 99 Solving -486 + 81x + 27x2 = 99 Solving for variable 'x'. Reorder the terms: -486 + -99 + 81x + 27x2 = 99 + -99 Combine like terms: -486 + -99 = -585 -585 + 81x + 27x2 = 99 + -99 Combine like terms: 99 + -99 = 0 -585 + 81x + 27x2 = 0 Factor out the Greatest Common Factor (GCF), '9'. 9(-65 + 9x + 3x2) = 0 Ignore the factor 9.Subproblem 1
Set the factor '(-65 + 9x + 3x2)' equal to zero and attempt to solve: Simplifying -65 + 9x + 3x2 = 0 Solving -65 + 9x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -21.66666667 + 3x + x2 = 0 Move the constant term to the right: Add '21.66666667' to each side of the equation. -21.66666667 + 3x + 21.66666667 + x2 = 0 + 21.66666667 Reorder the terms: -21.66666667 + 21.66666667 + 3x + x2 = 0 + 21.66666667 Combine like terms: -21.66666667 + 21.66666667 = 0.00000000 0.00000000 + 3x + x2 = 0 + 21.66666667 3x + x2 = 0 + 21.66666667 Combine like terms: 0 + 21.66666667 = 21.66666667 3x + x2 = 21.66666667 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 21.66666667 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 21.66666667 + 2.25 Combine like terms: 21.66666667 + 2.25 = 23.91666667 2.25 + 3x + x2 = 23.91666667 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 23.91666667 Calculate the square root of the right side: 4.890466917 Break this problem into two subproblems by setting (x + 1.5) equal to 4.890466917 and -4.890466917.Subproblem 1
x + 1.5 = 4.890466917 Simplifying x + 1.5 = 4.890466917 Reorder the terms: 1.5 + x = 4.890466917 Solving 1.5 + x = 4.890466917 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 4.890466917 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.890466917 + -1.5 x = 4.890466917 + -1.5 Combine like terms: 4.890466917 + -1.5 = 3.390466917 x = 3.390466917 Simplifying x = 3.390466917Subproblem 2
x + 1.5 = -4.890466917 Simplifying x + 1.5 = -4.890466917 Reorder the terms: 1.5 + x = -4.890466917 Solving 1.5 + x = -4.890466917 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -4.890466917 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.890466917 + -1.5 x = -4.890466917 + -1.5 Combine like terms: -4.890466917 + -1.5 = -6.390466917 x = -6.390466917 Simplifying x = -6.390466917Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.390466917, -6.390466917}Solution
x = {3.390466917, -6.390466917}
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