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Simplifying 2x(2 + 3x) = 45 (2 * 2x + 3x * 2x) = 45 (4x + 6x2) = 45 Solving 4x + 6x2 = 45 Solving for variable 'x'. Reorder the terms: -45 + 4x + 6x2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + 4x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -7.5 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 0.6666666667x + 7.5 + x2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 0.6666666667x + x2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 0.6666666667x + x2 = 0 + 7.5 0.6666666667x + x2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 0.6666666667x + x2 = 7.5 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 = 7.5 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 7.5 + 0.1111111112 Combine like terms: 7.5 + 0.1111111112 = 7.6111111112 0.1111111112 + 0.6666666667x + x2 = 7.6111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 7.6111111112 Calculate the square root of the right side: 2.758824226 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 2.758824226 and -2.758824226.Subproblem 1
x + 0.3333333334 = 2.758824226 Simplifying x + 0.3333333334 = 2.758824226 Reorder the terms: 0.3333333334 + x = 2.758824226 Solving 0.3333333334 + x = 2.758824226 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 = 2.758824226 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 2.758824226 + -0.3333333334 x = 2.758824226 + -0.3333333334 Combine like terms: 2.758824226 + -0.3333333334 = 2.4254908926 x = 2.4254908926 Simplifying x = 2.4254908926Subproblem 2
x + 0.3333333334 = -2.758824226 Simplifying x + 0.3333333334 = -2.758824226 Reorder the terms: 0.3333333334 + x = -2.758824226 Solving 0.3333333334 + x = -2.758824226 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 = -2.758824226 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -2.758824226 + -0.3333333334 x = -2.758824226 + -0.3333333334 Combine like terms: -2.758824226 + -0.3333333334 = -3.0921575594 x = -3.0921575594 Simplifying x = -3.0921575594Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.4254908926, -3.0921575594}
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