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Simplifying 0 = 12x2 + -80x + 84 Reorder the terms: 0 = 84 + -80x + 12x2 Solving 0 = 84 + -80x + 12x2 Solving for variable 'x'. Combine like terms: 0 + -84 = -84 -84 + 80x + -12x2 = 84 + -80x + 12x2 + -84 + 80x + -12x2 Reorder the terms: -84 + 80x + -12x2 = 84 + -84 + -80x + 80x + 12x2 + -12x2 Combine like terms: 84 + -84 = 0 -84 + 80x + -12x2 = 0 + -80x + 80x + 12x2 + -12x2 -84 + 80x + -12x2 = -80x + 80x + 12x2 + -12x2 Combine like terms: -80x + 80x = 0 -84 + 80x + -12x2 = 0 + 12x2 + -12x2 -84 + 80x + -12x2 = 12x2 + -12x2 Combine like terms: 12x2 + -12x2 = 0 -84 + 80x + -12x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-21 + 20x + -3x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-21 + 20x + -3x2)' equal to zero and attempt to solve: Simplifying -21 + 20x + -3x2 = 0 Solving -21 + 20x + -3x2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 7 + -6.666666667x + x2 = 0 Move the constant term to the right: Add '-7' to each side of the equation. 7 + -6.666666667x + -7 + x2 = 0 + -7 Reorder the terms: 7 + -7 + -6.666666667x + x2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + -6.666666667x + x2 = 0 + -7 -6.666666667x + x2 = 0 + -7 Combine like terms: 0 + -7 = -7 -6.666666667x + x2 = -7 The x term is -6.666666667x. Take half its coefficient (-3.333333334). Square it (11.11111112) and add it to both sides. Add '11.11111112' to each side of the equation. -6.666666667x + 11.11111112 + x2 = -7 + 11.11111112 Reorder the terms: 11.11111112 + -6.666666667x + x2 = -7 + 11.11111112 Combine like terms: -7 + 11.11111112 = 4.11111112 11.11111112 + -6.666666667x + x2 = 4.11111112 Factor a perfect square on the left side: (x + -3.333333334)(x + -3.333333334) = 4.11111112 Calculate the square root of the right side: 2.027587512 Break this problem into two subproblems by setting (x + -3.333333334) equal to 2.027587512 and -2.027587512.Subproblem 1
x + -3.333333334 = 2.027587512 Simplifying x + -3.333333334 = 2.027587512 Reorder the terms: -3.333333334 + x = 2.027587512 Solving -3.333333334 + x = 2.027587512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + x = 2.027587512 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + x = 2.027587512 + 3.333333334 x = 2.027587512 + 3.333333334 Combine like terms: 2.027587512 + 3.333333334 = 5.360920846 x = 5.360920846 Simplifying x = 5.360920846Subproblem 2
x + -3.333333334 = -2.027587512 Simplifying x + -3.333333334 = -2.027587512 Reorder the terms: -3.333333334 + x = -2.027587512 Solving -3.333333334 + x = -2.027587512 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.333333334' to each side of the equation. -3.333333334 + 3.333333334 + x = -2.027587512 + 3.333333334 Combine like terms: -3.333333334 + 3.333333334 = 0.000000000 0.000000000 + x = -2.027587512 + 3.333333334 x = -2.027587512 + 3.333333334 Combine like terms: -2.027587512 + 3.333333334 = 1.305745822 x = 1.305745822 Simplifying x = 1.305745822Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.360920846, 1.305745822}Solution
x = {5.360920846, 1.305745822}
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