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Simplifying (3x + 7)(6x + 3) = 62 Reorder the terms: (7 + 3x)(6x + 3) = 62 Reorder the terms: (7 + 3x)(3 + 6x) = 62 Multiply (7 + 3x) * (3 + 6x) (7(3 + 6x) + 3x * (3 + 6x)) = 62 ((3 * 7 + 6x * 7) + 3x * (3 + 6x)) = 62 ((21 + 42x) + 3x * (3 + 6x)) = 62 (21 + 42x + (3 * 3x + 6x * 3x)) = 62 (21 + 42x + (9x + 18x2)) = 62 Combine like terms: 42x + 9x = 51x (21 + 51x + 18x2) = 62 Solving 21 + 51x + 18x2 = 62 Solving for variable 'x'. Reorder the terms: 21 + -62 + 51x + 18x2 = 62 + -62 Combine like terms: 21 + -62 = -41 -41 + 51x + 18x2 = 62 + -62 Combine like terms: 62 + -62 = 0 -41 + 51x + 18x2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -2.277777778 + 2.833333333x + x2 = 0 Move the constant term to the right: Add '2.277777778' to each side of the equation. -2.277777778 + 2.833333333x + 2.277777778 + x2 = 0 + 2.277777778 Reorder the terms: -2.277777778 + 2.277777778 + 2.833333333x + x2 = 0 + 2.277777778 Combine like terms: -2.277777778 + 2.277777778 = 0.000000000 0.000000000 + 2.833333333x + x2 = 0 + 2.277777778 2.833333333x + x2 = 0 + 2.277777778 Combine like terms: 0 + 2.277777778 = 2.277777778 2.833333333x + x2 = 2.277777778 The x term is 2.833333333x. Take half its coefficient (1.416666667). Square it (2.006944445) and add it to both sides. Add '2.006944445' to each side of the equation. 2.833333333x + 2.006944445 + x2 = 2.277777778 + 2.006944445 Reorder the terms: 2.006944445 + 2.833333333x + x2 = 2.277777778 + 2.006944445 Combine like terms: 2.277777778 + 2.006944445 = 4.284722223 2.006944445 + 2.833333333x + x2 = 4.284722223 Factor a perfect square on the left side: (x + 1.416666667)(x + 1.416666667) = 4.284722223 Calculate the square root of the right side: 2.069957058 Break this problem into two subproblems by setting (x + 1.416666667) equal to 2.069957058 and -2.069957058.Subproblem 1
x + 1.416666667 = 2.069957058 Simplifying x + 1.416666667 = 2.069957058 Reorder the terms: 1.416666667 + x = 2.069957058 Solving 1.416666667 + x = 2.069957058 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = 2.069957058 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = 2.069957058 + -1.416666667 x = 2.069957058 + -1.416666667 Combine like terms: 2.069957058 + -1.416666667 = 0.653290391 x = 0.653290391 Simplifying x = 0.653290391Subproblem 2
x + 1.416666667 = -2.069957058 Simplifying x + 1.416666667 = -2.069957058 Reorder the terms: 1.416666667 + x = -2.069957058 Solving 1.416666667 + x = -2.069957058 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.416666667' to each side of the equation. 1.416666667 + -1.416666667 + x = -2.069957058 + -1.416666667 Combine like terms: 1.416666667 + -1.416666667 = 0.000000000 0.000000000 + x = -2.069957058 + -1.416666667 x = -2.069957058 + -1.416666667 Combine like terms: -2.069957058 + -1.416666667 = -3.486623725 x = -3.486623725 Simplifying x = -3.486623725Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.653290391, -3.486623725}
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