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Simplifying 24x2 + 24x + -3 = 0 Reorder the terms: -3 + 24x + 24x2 = 0 Solving -3 + 24x + 24x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '3'. 3(-1 + 8x + 8x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-1 + 8x + 8x2)' equal to zero and attempt to solve: Simplifying -1 + 8x + 8x2 = 0 Solving -1 + 8x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + x + x2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + x + 0.125 + x2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + x + x2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + x + x2 = 0 + 0.125 x + x2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 x + x2 = 0.125 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 0.125 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.125 + 0.25 Combine like terms: 0.125 + 0.25 = 0.375 1.25 + x2 = 0.375 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.375 Calculate the square root of the right side: 0.612372436 Break this problem into two subproblems by setting (x + 0.5) equal to 0.612372436 and -0.612372436.Subproblem 1
x + 0.5 = 0.612372436 Simplifying x + 0.5 = 0.612372436 Reorder the terms: 0.5 + x = 0.612372436 Solving 0.5 + x = 0.612372436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.612372436 + -0.5 x = 0.612372436 + -0.5 Combine like terms: 0.612372436 + -0.5 = 0.112372436 x = 0.112372436 Simplifying x = 0.112372436Subproblem 2
x + 0.5 = -0.612372436 Simplifying x + 0.5 = -0.612372436 Reorder the terms: 0.5 + x = -0.612372436 Solving 0.5 + x = -0.612372436 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.612372436 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.612372436 + -0.5 x = -0.612372436 + -0.5 Combine like terms: -0.612372436 + -0.5 = -1.112372436 x = -1.112372436 Simplifying x = -1.112372436Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.112372436, -1.112372436}Solution
x = {0.112372436, -1.112372436}
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