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Simplifying 3X(2X + 1) = 9 + -5(2X + -3) Reorder the terms: 3X(1 + 2X) = 9 + -5(2X + -3) (1 * 3X + 2X * 3X) = 9 + -5(2X + -3) (3X + 6X2) = 9 + -5(2X + -3) Reorder the terms: 3X + 6X2 = 9 + -5(-3 + 2X) 3X + 6X2 = 9 + (-3 * -5 + 2X * -5) 3X + 6X2 = 9 + (15 + -10X) Combine like terms: 9 + 15 = 24 3X + 6X2 = 24 + -10X Solving 3X + 6X2 = 24 + -10X Solving for variable 'X'. Reorder the terms: -24 + 3X + 10X + 6X2 = 24 + -10X + -24 + 10X Combine like terms: 3X + 10X = 13X -24 + 13X + 6X2 = 24 + -10X + -24 + 10X Reorder the terms: -24 + 13X + 6X2 = 24 + -24 + -10X + 10X Combine like terms: 24 + -24 = 0 -24 + 13X + 6X2 = 0 + -10X + 10X -24 + 13X + 6X2 = -10X + 10X Combine like terms: -10X + 10X = 0 -24 + 13X + 6X2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4 + 2.166666667X + X2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 2.166666667X + 4 + X2 = 0 + 4 Reorder the terms: -4 + 4 + 2.166666667X + X2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 2.166666667X + X2 = 0 + 4 2.166666667X + X2 = 0 + 4 Combine like terms: 0 + 4 = 4 2.166666667X + X2 = 4 The X term is 2.166666667X. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667X + 1.173611113 + X2 = 4 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667X + X2 = 4 + 1.173611113 Combine like terms: 4 + 1.173611113 = 5.173611113 1.173611113 + 2.166666667X + X2 = 5.173611113 Factor a perfect square on the left side: (X + 1.083333334)(X + 1.083333334) = 5.173611113 Calculate the square root of the right side: 2.274557344 Break this problem into two subproblems by setting (X + 1.083333334) equal to 2.274557344 and -2.274557344.Subproblem 1
X + 1.083333334 = 2.274557344 Simplifying X + 1.083333334 = 2.274557344 Reorder the terms: 1.083333334 + X = 2.274557344 Solving 1.083333334 + X = 2.274557344 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + X = 2.274557344 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + X = 2.274557344 + -1.083333334 X = 2.274557344 + -1.083333334 Combine like terms: 2.274557344 + -1.083333334 = 1.19122401 X = 1.19122401 Simplifying X = 1.19122401Subproblem 2
X + 1.083333334 = -2.274557344 Simplifying X + 1.083333334 = -2.274557344 Reorder the terms: 1.083333334 + X = -2.274557344 Solving 1.083333334 + X = -2.274557344 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + X = -2.274557344 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + X = -2.274557344 + -1.083333334 X = -2.274557344 + -1.083333334 Combine like terms: -2.274557344 + -1.083333334 = -3.357890678 X = -3.357890678 Simplifying X = -3.357890678Solution
The solution to the problem is based on the solutions from the subproblems. X = {1.19122401, -3.357890678}
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