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Simplifying -15x(2x + 7) = 14 Reorder the terms: -15x(7 + 2x) = 14 (7 * -15x + 2x * -15x) = 14 (-105x + -30x2) = 14 Solving -105x + -30x2 = 14 Solving for variable 'x'. Reorder the terms: -14 + -105x + -30x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + -105x + -30x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(14 + 105x + 30x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(14 + 105x + 30x2)' equal to zero and attempt to solve: Simplifying 14 + 105x + 30x2 = 0 Solving 14 + 105x + 30x2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. 0.4666666667 + 3.5x + x2 = 0.0 Move the constant term to the right: Add '-0.4666666667' to each side of the equation. 0.4666666667 + 3.5x + -0.4666666667 + x2 = 0.0 + -0.4666666667 Reorder the terms: 0.4666666667 + -0.4666666667 + 3.5x + x2 = 0.0 + -0.4666666667 Combine like terms: 0.4666666667 + -0.4666666667 = 0.0000000000 0.0000000000 + 3.5x + x2 = 0.0 + -0.4666666667 3.5x + x2 = 0.0 + -0.4666666667 Combine like terms: 0.0 + -0.4666666667 = -0.4666666667 3.5x + x2 = -0.4666666667 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = -0.4666666667 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = -0.4666666667 + 3.0625 Combine like terms: -0.4666666667 + 3.0625 = 2.5958333333 3.0625 + 3.5x + x2 = 2.5958333333 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 2.5958333333 Calculate the square root of the right side: 1.611159003 Break this problem into two subproblems by setting (x + 1.75) equal to 1.611159003 and -1.611159003.Subproblem 1
x + 1.75 = 1.611159003 Simplifying x + 1.75 = 1.611159003 Reorder the terms: 1.75 + x = 1.611159003 Solving 1.75 + x = 1.611159003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 1.611159003 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 1.611159003 + -1.75 x = 1.611159003 + -1.75 Combine like terms: 1.611159003 + -1.75 = -0.138840997 x = -0.138840997 Simplifying x = -0.138840997Subproblem 2
x + 1.75 = -1.611159003 Simplifying x + 1.75 = -1.611159003 Reorder the terms: 1.75 + x = -1.611159003 Solving 1.75 + x = -1.611159003 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -1.611159003 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -1.611159003 + -1.75 x = -1.611159003 + -1.75 Combine like terms: -1.611159003 + -1.75 = -3.361159003 x = -3.361159003 Simplifying x = -3.361159003Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.138840997, -3.361159003}Solution
x = {-0.138840997, -3.361159003}
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