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Simplifying 12 + 78x(x + 2) = 765 Reorder the terms: 12 + 78x(2 + x) = 765 12 + (2 * 78x + x * 78x) = 765 12 + (156x + 78x2) = 765 Solving 12 + 156x + 78x2 = 765 Solving for variable 'x'. Reorder the terms: 12 + -765 + 156x + 78x2 = 765 + -765 Combine like terms: 12 + -765 = -753 -753 + 156x + 78x2 = 765 + -765 Combine like terms: 765 + -765 = 0 -753 + 156x + 78x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-251 + 52x + 26x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-251 + 52x + 26x2)' equal to zero and attempt to solve: Simplifying -251 + 52x + 26x2 = 0 Solving -251 + 52x + 26x2 = 0 Begin completing the square. Divide all terms by 26 the coefficient of the squared term: Divide each side by '26'. -9.653846154 + 2x + x2 = 0 Move the constant term to the right: Add '9.653846154' to each side of the equation. -9.653846154 + 2x + 9.653846154 + x2 = 0 + 9.653846154 Reorder the terms: -9.653846154 + 9.653846154 + 2x + x2 = 0 + 9.653846154 Combine like terms: -9.653846154 + 9.653846154 = 0.000000000 0.000000000 + 2x + x2 = 0 + 9.653846154 2x + x2 = 0 + 9.653846154 Combine like terms: 0 + 9.653846154 = 9.653846154 2x + x2 = 9.653846154 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 9.653846154 + 1 Reorder the terms: 1 + 2x + x2 = 9.653846154 + 1 Combine like terms: 9.653846154 + 1 = 10.653846154 1 + 2x + x2 = 10.653846154 Factor a perfect square on the left side: (x + 1)(x + 1) = 10.653846154 Calculate the square root of the right side: 3.264023001 Break this problem into two subproblems by setting (x + 1) equal to 3.264023001 and -3.264023001.Subproblem 1
x + 1 = 3.264023001 Simplifying x + 1 = 3.264023001 Reorder the terms: 1 + x = 3.264023001 Solving 1 + x = 3.264023001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 3.264023001 + -1 Combine like terms: 1 + -1 = 0 0 + x = 3.264023001 + -1 x = 3.264023001 + -1 Combine like terms: 3.264023001 + -1 = 2.264023001 x = 2.264023001 Simplifying x = 2.264023001Subproblem 2
x + 1 = -3.264023001 Simplifying x + 1 = -3.264023001 Reorder the terms: 1 + x = -3.264023001 Solving 1 + x = -3.264023001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -3.264023001 + -1 Combine like terms: 1 + -1 = 0 0 + x = -3.264023001 + -1 x = -3.264023001 + -1 Combine like terms: -3.264023001 + -1 = -4.264023001 x = -4.264023001 Simplifying x = -4.264023001Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.264023001, -4.264023001}Solution
x = {2.264023001, -4.264023001}
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