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Simplifying 42x2 + -24x + -32 = 0 Reorder the terms: -32 + -24x + 42x2 = 0 Solving -32 + -24x + 42x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-16 + -12x + 21x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-16 + -12x + 21x2)' equal to zero and attempt to solve: Simplifying -16 + -12x + 21x2 = 0 Solving -16 + -12x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.7619047619 + -0.5714285714x + x2 = 0 Move the constant term to the right: Add '0.7619047619' to each side of the equation. -0.7619047619 + -0.5714285714x + 0.7619047619 + x2 = 0 + 0.7619047619 Reorder the terms: -0.7619047619 + 0.7619047619 + -0.5714285714x + x2 = 0 + 0.7619047619 Combine like terms: -0.7619047619 + 0.7619047619 = 0.0000000000 0.0000000000 + -0.5714285714x + x2 = 0 + 0.7619047619 -0.5714285714x + x2 = 0 + 0.7619047619 Combine like terms: 0 + 0.7619047619 = 0.7619047619 -0.5714285714x + x2 = 0.7619047619 The x term is -0.5714285714x. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714x + 0.08163265305 + x2 = 0.7619047619 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714x + x2 = 0.7619047619 + 0.08163265305 Combine like terms: 0.7619047619 + 0.08163265305 = 0.84353741495 0.08163265305 + -0.5714285714x + x2 = 0.84353741495 Factor a perfect square on the left side: (x + -0.2857142857)(x + -0.2857142857) = 0.84353741495 Calculate the square root of the right side: 0.91844293 Break this problem into two subproblems by setting (x + -0.2857142857) equal to 0.91844293 and -0.91844293.Subproblem 1
x + -0.2857142857 = 0.91844293 Simplifying x + -0.2857142857 = 0.91844293 Reorder the terms: -0.2857142857 + x = 0.91844293 Solving -0.2857142857 + x = 0.91844293 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = 0.91844293 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = 0.91844293 + 0.2857142857 x = 0.91844293 + 0.2857142857 Combine like terms: 0.91844293 + 0.2857142857 = 1.2041572157 x = 1.2041572157 Simplifying x = 1.2041572157Subproblem 2
x + -0.2857142857 = -0.91844293 Simplifying x + -0.2857142857 = -0.91844293 Reorder the terms: -0.2857142857 + x = -0.91844293 Solving -0.2857142857 + x = -0.91844293 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = -0.91844293 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = -0.91844293 + 0.2857142857 x = -0.91844293 + 0.2857142857 Combine like terms: -0.91844293 + 0.2857142857 = -0.6327286443 x = -0.6327286443 Simplifying x = -0.6327286443Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.2041572157, -0.6327286443}Solution
x = {1.2041572157, -0.6327286443}
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