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Simplifying (13x + 3)(9x + 1) = 180 Reorder the terms: (3 + 13x)(9x + 1) = 180 Reorder the terms: (3 + 13x)(1 + 9x) = 180 Multiply (3 + 13x) * (1 + 9x) (3(1 + 9x) + 13x * (1 + 9x)) = 180 ((1 * 3 + 9x * 3) + 13x * (1 + 9x)) = 180 ((3 + 27x) + 13x * (1 + 9x)) = 180 (3 + 27x + (1 * 13x + 9x * 13x)) = 180 (3 + 27x + (13x + 117x2)) = 180 Combine like terms: 27x + 13x = 40x (3 + 40x + 117x2) = 180 Solving 3 + 40x + 117x2 = 180 Solving for variable 'x'. Reorder the terms: 3 + -180 + 40x + 117x2 = 180 + -180 Combine like terms: 3 + -180 = -177 -177 + 40x + 117x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -177 + 40x + 117x2 = 0 Begin completing the square. Divide all terms by 117 the coefficient of the squared term: Divide each side by '117'. -1.512820513 + 0.3418803419x + x2 = 0 Move the constant term to the right: Add '1.512820513' to each side of the equation. -1.512820513 + 0.3418803419x + 1.512820513 + x2 = 0 + 1.512820513 Reorder the terms: -1.512820513 + 1.512820513 + 0.3418803419x + x2 = 0 + 1.512820513 Combine like terms: -1.512820513 + 1.512820513 = 0.000000000 0.000000000 + 0.3418803419x + x2 = 0 + 1.512820513 0.3418803419x + x2 = 0 + 1.512820513 Combine like terms: 0 + 1.512820513 = 1.512820513 0.3418803419x + x2 = 1.512820513 The x term is 0.3418803419x. Take half its coefficient (0.170940171). Square it (0.02922054206) and add it to both sides. Add '0.02922054206' to each side of the equation. 0.3418803419x + 0.02922054206 + x2 = 1.512820513 + 0.02922054206 Reorder the terms: 0.02922054206 + 0.3418803419x + x2 = 1.512820513 + 0.02922054206 Combine like terms: 1.512820513 + 0.02922054206 = 1.54204105506 0.02922054206 + 0.3418803419x + x2 = 1.54204105506 Factor a perfect square on the left side: (x + 0.170940171)(x + 0.170940171) = 1.54204105506 Calculate the square root of the right side: 1.241789457 Break this problem into two subproblems by setting (x + 0.170940171) equal to 1.241789457 and -1.241789457.Subproblem 1
x + 0.170940171 = 1.241789457 Simplifying x + 0.170940171 = 1.241789457 Reorder the terms: 0.170940171 + x = 1.241789457 Solving 0.170940171 + x = 1.241789457 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.170940171' to each side of the equation. 0.170940171 + -0.170940171 + x = 1.241789457 + -0.170940171 Combine like terms: 0.170940171 + -0.170940171 = 0.000000000 0.000000000 + x = 1.241789457 + -0.170940171 x = 1.241789457 + -0.170940171 Combine like terms: 1.241789457 + -0.170940171 = 1.070849286 x = 1.070849286 Simplifying x = 1.070849286Subproblem 2
x + 0.170940171 = -1.241789457 Simplifying x + 0.170940171 = -1.241789457 Reorder the terms: 0.170940171 + x = -1.241789457 Solving 0.170940171 + x = -1.241789457 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.170940171' to each side of the equation. 0.170940171 + -0.170940171 + x = -1.241789457 + -0.170940171 Combine like terms: 0.170940171 + -0.170940171 = 0.000000000 0.000000000 + x = -1.241789457 + -0.170940171 x = -1.241789457 + -0.170940171 Combine like terms: -1.241789457 + -0.170940171 = -1.412729628 x = -1.412729628 Simplifying x = -1.412729628Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.070849286, -1.412729628}
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