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Simplifying 4p + 25 = 6p(p + -3) + -3(4 + -3p) Reorder the terms: 25 + 4p = 6p(p + -3) + -3(4 + -3p) Reorder the terms: 25 + 4p = 6p(-3 + p) + -3(4 + -3p) 25 + 4p = (-3 * 6p + p * 6p) + -3(4 + -3p) 25 + 4p = (-18p + 6p2) + -3(4 + -3p) 25 + 4p = -18p + 6p2 + (4 * -3 + -3p * -3) 25 + 4p = -18p + 6p2 + (-12 + 9p) Reorder the terms: 25 + 4p = -12 + -18p + 9p + 6p2 Combine like terms: -18p + 9p = -9p 25 + 4p = -12 + -9p + 6p2 Solving 25 + 4p = -12 + -9p + 6p2 Solving for variable 'p'. Reorder the terms: 25 + 12 + 4p + 9p + -6p2 = -12 + -9p + 6p2 + 12 + 9p + -6p2 Combine like terms: 25 + 12 = 37 37 + 4p + 9p + -6p2 = -12 + -9p + 6p2 + 12 + 9p + -6p2 Combine like terms: 4p + 9p = 13p 37 + 13p + -6p2 = -12 + -9p + 6p2 + 12 + 9p + -6p2 Reorder the terms: 37 + 13p + -6p2 = -12 + 12 + -9p + 9p + 6p2 + -6p2 Combine like terms: -12 + 12 = 0 37 + 13p + -6p2 = 0 + -9p + 9p + 6p2 + -6p2 37 + 13p + -6p2 = -9p + 9p + 6p2 + -6p2 Combine like terms: -9p + 9p = 0 37 + 13p + -6p2 = 0 + 6p2 + -6p2 37 + 13p + -6p2 = 6p2 + -6p2 Combine like terms: 6p2 + -6p2 = 0 37 + 13p + -6p2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -6.166666667 + -2.166666667p + p2 = 0 Move the constant term to the right: Add '6.166666667' to each side of the equation. -6.166666667 + -2.166666667p + 6.166666667 + p2 = 0 + 6.166666667 Reorder the terms: -6.166666667 + 6.166666667 + -2.166666667p + p2 = 0 + 6.166666667 Combine like terms: -6.166666667 + 6.166666667 = 0.000000000 0.000000000 + -2.166666667p + p2 = 0 + 6.166666667 -2.166666667p + p2 = 0 + 6.166666667 Combine like terms: 0 + 6.166666667 = 6.166666667 -2.166666667p + p2 = 6.166666667 The p term is -2.166666667p. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667p + 1.173611113 + p2 = 6.166666667 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667p + p2 = 6.166666667 + 1.173611113 Combine like terms: 6.166666667 + 1.173611113 = 7.34027778 1.173611113 + -2.166666667p + p2 = 7.34027778 Factor a perfect square on the left side: (p + -1.083333334)(p + -1.083333334) = 7.34027778 Calculate the square root of the right side: 2.709294702 Break this problem into two subproblems by setting (p + -1.083333334) equal to 2.709294702 and -2.709294702.Subproblem 1
p + -1.083333334 = 2.709294702 Simplifying p + -1.083333334 = 2.709294702 Reorder the terms: -1.083333334 + p = 2.709294702 Solving -1.083333334 + p = 2.709294702 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + p = 2.709294702 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + p = 2.709294702 + 1.083333334 p = 2.709294702 + 1.083333334 Combine like terms: 2.709294702 + 1.083333334 = 3.792628036 p = 3.792628036 Simplifying p = 3.792628036Subproblem 2
p + -1.083333334 = -2.709294702 Simplifying p + -1.083333334 = -2.709294702 Reorder the terms: -1.083333334 + p = -2.709294702 Solving -1.083333334 + p = -2.709294702 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + p = -2.709294702 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + p = -2.709294702 + 1.083333334 p = -2.709294702 + 1.083333334 Combine like terms: -2.709294702 + 1.083333334 = -1.625961368 p = -1.625961368 Simplifying p = -1.625961368Solution
The solution to the problem is based on the solutions from the subproblems. p = {3.792628036, -1.625961368}
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