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Simplifying 4x(7x + 12) = 14 Reorder the terms: 4x(12 + 7x) = 14 (12 * 4x + 7x * 4x) = 14 (48x + 28x2) = 14 Solving 48x + 28x2 = 14 Solving for variable 'x'. Reorder the terms: -14 + 48x + 28x2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + 48x + 28x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 24x + 14x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 24x + 14x2)' equal to zero and attempt to solve: Simplifying -7 + 24x + 14x2 = 0 Solving -7 + 24x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.5 + 1.714285714x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.714285714x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.714285714x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.714285714x + x2 = 0 + 0.5 1.714285714x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.714285714x + x2 = 0.5 The x term is 1.714285714x. Take half its coefficient (0.857142857). Square it (0.7346938773) and add it to both sides. Add '0.7346938773' to each side of the equation. 1.714285714x + 0.7346938773 + x2 = 0.5 + 0.7346938773 Reorder the terms: 0.7346938773 + 1.714285714x + x2 = 0.5 + 0.7346938773 Combine like terms: 0.5 + 0.7346938773 = 1.2346938773 0.7346938773 + 1.714285714x + x2 = 1.2346938773 Factor a perfect square on the left side: (x + 0.857142857)(x + 0.857142857) = 1.2346938773 Calculate the square root of the right side: 1.111167799 Break this problem into two subproblems by setting (x + 0.857142857) equal to 1.111167799 and -1.111167799.Subproblem 1
x + 0.857142857 = 1.111167799 Simplifying x + 0.857142857 = 1.111167799 Reorder the terms: 0.857142857 + x = 1.111167799 Solving 0.857142857 + x = 1.111167799 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = 1.111167799 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = 1.111167799 + -0.857142857 x = 1.111167799 + -0.857142857 Combine like terms: 1.111167799 + -0.857142857 = 0.254024942 x = 0.254024942 Simplifying x = 0.254024942Subproblem 2
x + 0.857142857 = -1.111167799 Simplifying x + 0.857142857 = -1.111167799 Reorder the terms: 0.857142857 + x = -1.111167799 Solving 0.857142857 + x = -1.111167799 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.857142857' to each side of the equation. 0.857142857 + -0.857142857 + x = -1.111167799 + -0.857142857 Combine like terms: 0.857142857 + -0.857142857 = 0.000000000 0.000000000 + x = -1.111167799 + -0.857142857 x = -1.111167799 + -0.857142857 Combine like terms: -1.111167799 + -0.857142857 = -1.968310656 x = -1.968310656 Simplifying x = -1.968310656Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.254024942, -1.968310656}Solution
x = {0.254024942, -1.968310656}
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