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Simplifying 8x(7x + 15) = 180 Reorder the terms: 8x(15 + 7x) = 180 (15 * 8x + 7x * 8x) = 180 (120x + 56x2) = 180 Solving 120x + 56x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 120x + 56x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 120x + 56x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-45 + 30x + 14x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-45 + 30x + 14x2)' equal to zero and attempt to solve: Simplifying -45 + 30x + 14x2 = 0 Solving -45 + 30x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -3.214285714 + 2.142857143x + x2 = 0 Move the constant term to the right: Add '3.214285714' to each side of the equation. -3.214285714 + 2.142857143x + 3.214285714 + x2 = 0 + 3.214285714 Reorder the terms: -3.214285714 + 3.214285714 + 2.142857143x + x2 = 0 + 3.214285714 Combine like terms: -3.214285714 + 3.214285714 = 0.000000000 0.000000000 + 2.142857143x + x2 = 0 + 3.214285714 2.142857143x + x2 = 0 + 3.214285714 Combine like terms: 0 + 3.214285714 = 3.214285714 2.142857143x + x2 = 3.214285714 The x term is 2.142857143x. Take half its coefficient (1.071428572). Square it (1.147959185) and add it to both sides. Add '1.147959185' to each side of the equation. 2.142857143x + 1.147959185 + x2 = 3.214285714 + 1.147959185 Reorder the terms: 1.147959185 + 2.142857143x + x2 = 3.214285714 + 1.147959185 Combine like terms: 3.214285714 + 1.147959185 = 4.362244899 1.147959185 + 2.142857143x + x2 = 4.362244899 Factor a perfect square on the left side: (x + 1.071428572)(x + 1.071428572) = 4.362244899 Calculate the square root of the right side: 2.088598788 Break this problem into two subproblems by setting (x + 1.071428572) equal to 2.088598788 and -2.088598788.Subproblem 1
x + 1.071428572 = 2.088598788 Simplifying x + 1.071428572 = 2.088598788 Reorder the terms: 1.071428572 + x = 2.088598788 Solving 1.071428572 + x = 2.088598788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + x = 2.088598788 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + x = 2.088598788 + -1.071428572 x = 2.088598788 + -1.071428572 Combine like terms: 2.088598788 + -1.071428572 = 1.017170216 x = 1.017170216 Simplifying x = 1.017170216Subproblem 2
x + 1.071428572 = -2.088598788 Simplifying x + 1.071428572 = -2.088598788 Reorder the terms: 1.071428572 + x = -2.088598788 Solving 1.071428572 + x = -2.088598788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.071428572' to each side of the equation. 1.071428572 + -1.071428572 + x = -2.088598788 + -1.071428572 Combine like terms: 1.071428572 + -1.071428572 = 0.000000000 0.000000000 + x = -2.088598788 + -1.071428572 x = -2.088598788 + -1.071428572 Combine like terms: -2.088598788 + -1.071428572 = -3.16002736 x = -3.16002736 Simplifying x = -3.16002736Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.017170216, -3.16002736}Solution
x = {1.017170216, -3.16002736}
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