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Simplifying (2x + 3)(3x + 7) = 35 Reorder the terms: (3 + 2x)(3x + 7) = 35 Reorder the terms: (3 + 2x)(7 + 3x) = 35 Multiply (3 + 2x) * (7 + 3x) (3(7 + 3x) + 2x * (7 + 3x)) = 35 ((7 * 3 + 3x * 3) + 2x * (7 + 3x)) = 35 ((21 + 9x) + 2x * (7 + 3x)) = 35 (21 + 9x + (7 * 2x + 3x * 2x)) = 35 (21 + 9x + (14x + 6x2)) = 35 Combine like terms: 9x + 14x = 23x (21 + 23x + 6x2) = 35 Solving 21 + 23x + 6x2 = 35 Solving for variable 'x'. Reorder the terms: 21 + -35 + 23x + 6x2 = 35 + -35 Combine like terms: 21 + -35 = -14 -14 + 23x + 6x2 = 35 + -35 Combine like terms: 35 + -35 = 0 -14 + 23x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.333333333 + 3.833333333x + x2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + 3.833333333x + 2.333333333 + x2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + 3.833333333x + x2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + 3.833333333x + x2 = 0 + 2.333333333 3.833333333x + x2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 3.833333333x + x2 = 2.333333333 The x term is 3.833333333x. Take half its coefficient (1.916666667). Square it (3.673611112) and add it to both sides. Add '3.673611112' to each side of the equation. 3.833333333x + 3.673611112 + x2 = 2.333333333 + 3.673611112 Reorder the terms: 3.673611112 + 3.833333333x + x2 = 2.333333333 + 3.673611112 Combine like terms: 2.333333333 + 3.673611112 = 6.006944445 3.673611112 + 3.833333333x + x2 = 6.006944445 Factor a perfect square on the left side: (x + 1.916666667)(x + 1.916666667) = 6.006944445 Calculate the square root of the right side: 2.450906862 Break this problem into two subproblems by setting (x + 1.916666667) equal to 2.450906862 and -2.450906862.Subproblem 1
x + 1.916666667 = 2.450906862 Simplifying x + 1.916666667 = 2.450906862 Reorder the terms: 1.916666667 + x = 2.450906862 Solving 1.916666667 + x = 2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = 2.450906862 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = 2.450906862 + -1.916666667 x = 2.450906862 + -1.916666667 Combine like terms: 2.450906862 + -1.916666667 = 0.534240195 x = 0.534240195 Simplifying x = 0.534240195Subproblem 2
x + 1.916666667 = -2.450906862 Simplifying x + 1.916666667 = -2.450906862 Reorder the terms: 1.916666667 + x = -2.450906862 Solving 1.916666667 + x = -2.450906862 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.916666667' to each side of the equation. 1.916666667 + -1.916666667 + x = -2.450906862 + -1.916666667 Combine like terms: 1.916666667 + -1.916666667 = 0.000000000 0.000000000 + x = -2.450906862 + -1.916666667 x = -2.450906862 + -1.916666667 Combine like terms: -2.450906862 + -1.916666667 = -4.367573529 x = -4.367573529 Simplifying x = -4.367573529Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.534240195, -4.367573529}
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