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Simplifying 3x2 + 2x + 45 = 180 Reorder the terms: 45 + 2x + 3x2 = 180 Solving 45 + 2x + 3x2 = 180 Solving for variable 'x'. Reorder the terms: 45 + -180 + 2x + 3x2 = 180 + -180 Combine like terms: 45 + -180 = -135 -135 + 2x + 3x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -135 + 2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -45 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '45' to each side of the equation. -45 + 0.6666666667x + 45 + x2 = 0 + 45 Reorder the terms: -45 + 45 + 0.6666666667x + x2 = 0 + 45 Combine like terms: -45 + 45 = 0 0 + 0.6666666667x + x2 = 0 + 45 0.6666666667x + x2 = 0 + 45 Combine like terms: 0 + 45 = 45 0.6666666667x + x2 = 45 The x term is 0.6666666667x. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667x + 0.1111111112 + x2 = 45 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 45 + 0.1111111112 Combine like terms: 45 + 0.1111111112 = 45.1111111112 0.1111111112 + 0.6666666667x + x2 = 45.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 45.1111111112 Calculate the square root of the right side: 6.71648056 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 6.71648056 and -6.71648056.Subproblem 1
x + 0.3333333334 = 6.71648056 Simplifying x + 0.3333333334 = 6.71648056 Reorder the terms: 0.3333333334 + x = 6.71648056 Solving 0.3333333334 + x = 6.71648056 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = 6.71648056 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 6.71648056 + -0.3333333334 x = 6.71648056 + -0.3333333334 Combine like terms: 6.71648056 + -0.3333333334 = 6.3831472266 x = 6.3831472266 Simplifying x = 6.3831472266Subproblem 2
x + 0.3333333334 = -6.71648056 Simplifying x + 0.3333333334 = -6.71648056 Reorder the terms: 0.3333333334 + x = -6.71648056 Solving 0.3333333334 + x = -6.71648056 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + x = -6.71648056 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -6.71648056 + -0.3333333334 x = -6.71648056 + -0.3333333334 Combine like terms: -6.71648056 + -0.3333333334 = -7.0498138934 x = -7.0498138934 Simplifying x = -7.0498138934Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.3831472266, -7.0498138934}
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