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Simplifying -42p2 + -32p + 14 = 0 Reorder the terms: 14 + -32p + -42p2 = 0 Solving 14 + -32p + -42p2 = 0 Solving for variable 'p'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -16p + -21p2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -16p + -21p2)' equal to zero and attempt to solve: Simplifying 7 + -16p + -21p2 = 0 Solving 7 + -16p + -21p2 = 0 Begin completing the square. Divide all terms by -21 the coefficient of the squared term: Divide each side by '-21'. -0.3333333333 + 0.7619047619p + p2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 0.7619047619p + 0.3333333333 + p2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 0.7619047619p + p2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 0.7619047619p + p2 = 0 + 0.3333333333 0.7619047619p + p2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 0.7619047619p + p2 = 0.3333333333 The p term is 0.7619047619p. Take half its coefficient (0.380952381). Square it (0.1451247166) and add it to both sides. Add '0.1451247166' to each side of the equation. 0.7619047619p + 0.1451247166 + p2 = 0.3333333333 + 0.1451247166 Reorder the terms: 0.1451247166 + 0.7619047619p + p2 = 0.3333333333 + 0.1451247166 Combine like terms: 0.3333333333 + 0.1451247166 = 0.4784580499 0.1451247166 + 0.7619047619p + p2 = 0.4784580499 Factor a perfect square on the left side: (p + 0.380952381)(p + 0.380952381) = 0.4784580499 Calculate the square root of the right side: 0.691706621 Break this problem into two subproblems by setting (p + 0.380952381) equal to 0.691706621 and -0.691706621.Subproblem 1
p + 0.380952381 = 0.691706621 Simplifying p + 0.380952381 = 0.691706621 Reorder the terms: 0.380952381 + p = 0.691706621 Solving 0.380952381 + p = 0.691706621 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.380952381' to each side of the equation. 0.380952381 + -0.380952381 + p = 0.691706621 + -0.380952381 Combine like terms: 0.380952381 + -0.380952381 = 0.000000000 0.000000000 + p = 0.691706621 + -0.380952381 p = 0.691706621 + -0.380952381 Combine like terms: 0.691706621 + -0.380952381 = 0.31075424 p = 0.31075424 Simplifying p = 0.31075424Subproblem 2
p + 0.380952381 = -0.691706621 Simplifying p + 0.380952381 = -0.691706621 Reorder the terms: 0.380952381 + p = -0.691706621 Solving 0.380952381 + p = -0.691706621 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.380952381' to each side of the equation. 0.380952381 + -0.380952381 + p = -0.691706621 + -0.380952381 Combine like terms: 0.380952381 + -0.380952381 = 0.000000000 0.000000000 + p = -0.691706621 + -0.380952381 p = -0.691706621 + -0.380952381 Combine like terms: -0.691706621 + -0.380952381 = -1.072659002 p = -1.072659002 Simplifying p = -1.072659002Solution
The solution to the problem is based on the solutions from the subproblems. p = {0.31075424, -1.072659002}Solution
p = {0.31075424, -1.072659002}
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