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Simplifying 72 = 0.5(3x + 6)(x + 4) Reorder the terms: 72 = 0.5(6 + 3x)(x + 4) Reorder the terms: 72 = 0.5(6 + 3x)(4 + x) Multiply (6 + 3x) * (4 + x) 72 = 0.5(6(4 + x) + 3x * (4 + x)) 72 = 0.5((4 * 6 + x * 6) + 3x * (4 + x)) 72 = 0.5((24 + 6x) + 3x * (4 + x)) 72 = 0.5(24 + 6x + (4 * 3x + x * 3x)) 72 = 0.5(24 + 6x + (12x + 3x2)) Combine like terms: 6x + 12x = 18x 72 = 0.5(24 + 18x + 3x2) 72 = (24 * 0.5 + 18x * 0.5 + 3x2 * 0.5) 72 = (12 + 9x + 1.5x2) Solving 72 = 12 + 9x + 1.5x2 Solving for variable 'x'. Combine like terms: 72 + -12 = 60 60 + -9x + -1.5x2 = 12 + 9x + 1.5x2 + -12 + -9x + -1.5x2 Reorder the terms: 60 + -9x + -1.5x2 = 12 + -12 + 9x + -9x + 1.5x2 + -1.5x2 Combine like terms: 12 + -12 = 0 60 + -9x + -1.5x2 = 0 + 9x + -9x + 1.5x2 + -1.5x2 60 + -9x + -1.5x2 = 9x + -9x + 1.5x2 + -1.5x2 Combine like terms: 9x + -9x = 0 60 + -9x + -1.5x2 = 0 + 1.5x2 + -1.5x2 60 + -9x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 60 + -9x + -1.5x2 = 0.0 Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. -40 + 6x + x2 = 0 Move the constant term to the right: Add '40' to each side of the equation. -40 + 6x + 40 + x2 = 0 + 40 Reorder the terms: -40 + 40 + 6x + x2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 6x + x2 = 0 + 40 6x + x2 = 0 + 40 Combine like terms: 0 + 40 = 40 6x + x2 = 40 The x term is 6x. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6x + 9 + x2 = 40 + 9 Reorder the terms: 9 + 6x + x2 = 40 + 9 Combine like terms: 40 + 9 = 49 9 + 6x + x2 = 49 Factor a perfect square on the left side: (x + 3)(x + 3) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (x + 3) equal to 7 and -7.Subproblem 1
x + 3 = 7 Simplifying x + 3 = 7 Reorder the terms: 3 + x = 7 Solving 3 + x = 7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 7 + -3 Combine like terms: 3 + -3 = 0 0 + x = 7 + -3 x = 7 + -3 Combine like terms: 7 + -3 = 4 x = 4 Simplifying x = 4Subproblem 2
x + 3 = -7 Simplifying x + 3 = -7 Reorder the terms: 3 + x = -7 Solving 3 + x = -7 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -7 + -3 Combine like terms: 3 + -3 = 0 0 + x = -7 + -3 x = -7 + -3 Combine like terms: -7 + -3 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {4, -10}
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