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Simplifying (x)(3x + 2) = 420 Reorder the terms: x(2 + 3x) = 420 (2 * x + 3x * x) = 420 (2x + 3x2) = 420 Solving 2x + 3x2 = 420 Solving for variable 'x'. Reorder the terms: -420 + 2x + 3x2 = 420 + -420 Combine like terms: 420 + -420 = 0 -420 + 2x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -140 + 0.6666666667x + x2 = 0 Move the constant term to the right: Add '140' to each side of the equation. -140 + 0.6666666667x + 140 + x2 = 0 + 140 Reorder the terms: -140 + 140 + 0.6666666667x + x2 = 0 + 140 Combine like terms: -140 + 140 = 0 0 + 0.6666666667x + x2 = 0 + 140 0.6666666667x + x2 = 0 + 140 Combine like terms: 0 + 140 = 140 0.6666666667x + x2 = 140 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 = 140 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667x + x2 = 140 + 0.1111111112 Combine like terms: 140 + 0.1111111112 = 140.1111111112 0.1111111112 + 0.6666666667x + x2 = 140.1111111112 Factor a perfect square on the left side: (x + 0.3333333334)(x + 0.3333333334) = 140.1111111112 Calculate the square root of the right side: 11.836853936 Break this problem into two subproblems by setting (x + 0.3333333334) equal to 11.836853936 and -11.836853936.Subproblem 1
x + 0.3333333334 = 11.836853936 Simplifying x + 0.3333333334 = 11.836853936 Reorder the terms: 0.3333333334 + x = 11.836853936 Solving 0.3333333334 + x = 11.836853936 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 = 11.836853936 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = 11.836853936 + -0.3333333334 x = 11.836853936 + -0.3333333334 Combine like terms: 11.836853936 + -0.3333333334 = 11.5035206026 x = 11.5035206026 Simplifying x = 11.5035206026Subproblem 2
x + 0.3333333334 = -11.836853936 Simplifying x + 0.3333333334 = -11.836853936 Reorder the terms: 0.3333333334 + x = -11.836853936 Solving 0.3333333334 + x = -11.836853936 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 = -11.836853936 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + x = -11.836853936 + -0.3333333334 x = -11.836853936 + -0.3333333334 Combine like terms: -11.836853936 + -0.3333333334 = -12.1701872694 x = -12.1701872694 Simplifying x = -12.1701872694Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.5035206026, -12.1701872694}
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