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Simplifying 324 = (6 + x)(3 + x) Multiply (6 + x) * (3 + x) 324 = (6(3 + x) + x(3 + x)) 324 = ((3 * 6 + x * 6) + x(3 + x)) 324 = ((18 + 6x) + x(3 + x)) 324 = (18 + 6x + (3 * x + x * x)) 324 = (18 + 6x + (3x + x2)) Combine like terms: 6x + 3x = 9x 324 = (18 + 9x + x2) Solving 324 = 18 + 9x + x2 Solving for variable 'x'. Combine like terms: 324 + -18 = 306 306 + -9x + -1x2 = 18 + 9x + x2 + -18 + -9x + -1x2 Reorder the terms: 306 + -9x + -1x2 = 18 + -18 + 9x + -9x + x2 + -1x2 Combine like terms: 18 + -18 = 0 306 + -9x + -1x2 = 0 + 9x + -9x + x2 + -1x2 306 + -9x + -1x2 = 9x + -9x + x2 + -1x2 Combine like terms: 9x + -9x = 0 306 + -9x + -1x2 = 0 + x2 + -1x2 306 + -9x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 306 + -9x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -306 + 9x + x2 = 0 Move the constant term to the right: Add '306' to each side of the equation. -306 + 9x + 306 + x2 = 0 + 306 Reorder the terms: -306 + 306 + 9x + x2 = 0 + 306 Combine like terms: -306 + 306 = 0 0 + 9x + x2 = 0 + 306 9x + x2 = 0 + 306 Combine like terms: 0 + 306 = 306 9x + x2 = 306 The x term is 9x. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9x + 20.25 + x2 = 306 + 20.25 Reorder the terms: 20.25 + 9x + x2 = 306 + 20.25 Combine like terms: 306 + 20.25 = 326.25 20.25 + 9x + x2 = 326.25 Factor a perfect square on the left side: (x + 4.5)(x + 4.5) = 326.25 Calculate the square root of the right side: 18.062391868 Break this problem into two subproblems by setting (x + 4.5) equal to 18.062391868 and -18.062391868.Subproblem 1
x + 4.5 = 18.062391868 Simplifying x + 4.5 = 18.062391868 Reorder the terms: 4.5 + x = 18.062391868 Solving 4.5 + x = 18.062391868 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = 18.062391868 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = 18.062391868 + -4.5 x = 18.062391868 + -4.5 Combine like terms: 18.062391868 + -4.5 = 13.562391868 x = 13.562391868 Simplifying x = 13.562391868Subproblem 2
x + 4.5 = -18.062391868 Simplifying x + 4.5 = -18.062391868 Reorder the terms: 4.5 + x = -18.062391868 Solving 4.5 + x = -18.062391868 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = -18.062391868 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = -18.062391868 + -4.5 x = -18.062391868 + -4.5 Combine like terms: -18.062391868 + -4.5 = -22.562391868 x = -22.562391868 Simplifying x = -22.562391868Solution
The solution to the problem is based on the solutions from the subproblems. x = {13.562391868, -22.562391868}
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